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\begin{verbatim}#|

 Copyright (C) 1994 by Misao Nagayama and Carolyn Talcott.  All Rights Reserved.

 You may copy and distribute verbatim copies of this Nqthm-1992 event script as
 you receive it, in any medium, including embedding it verbatim in derivative
 works, provided that you conspicuously and appropriately publish on each copy
 a valid copyright notice "Copyright (C) 1994 by Misao Nagayama and Carolyn
 Talcott.  All Rights Reserved."

 NO WARRANTY

 Misao Nagayama and Carolyn Talcott PROVIDE ABSOLUTELY NO WARRANTY.  THE EVENT
 SCRIPT IS PROVIDED "AS IS" WITHOUT WARRANTY OF ANY KIND, EITHER EXPRESS OR
 IMPLIED, INCLUDING, BUT NOT LIMITED TO, ANY IMPLIED WARRANTIES OF
 MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.  THE ENTIRE RISK AS TO
 THE QUALITY AND PERFORMANCE OF THE SCRIPT IS WITH YOU.  SHOULD THE SCRIPT
 PROVE DEFECTIVE, YOU ASSUME THE COST OF ALL NECESSARY SERVICING, REPAIR OR
 CORRECTION.

 IN NO EVENT WILL Misao Nagayama or Carolyn Talcott BE LIABLE TO YOU FOR ANY
 DAMAGES, ANY LOST PROFITS, LOST MONIES, OR OTHER SPECIAL, INCIDENTAL OR
 CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THIS SCRIPT
 (INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INACCURATE
 OR LOSSES SUSTAINED BY THIRD PARTIES), EVEN IF YOU HAVE ADVISED US OF THE
 POSSIBILITY OF SUCH DAMAGES, OR FOR ANY CLAIM BY ANY OTHER PARTY.

|#

\end{verbatim}
\noindent{\sc Event}:   Start with the initial {\bf nqthm} theory.

\addvspace{18pt}

\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;   mutex-molecular.ev    ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;   com.ev   ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;*sequence and finite set utilities

;;;The ith entry in l.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{nth} \\  
\=\+{\rm{nth}}\,({\it{l\/}}, {\it{i\/}})\index{nth} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{l\/}}) \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\rm{car}}\,({\it{l\/}}) \\ 
{\bf else }{\rm{nth}}\,({\rm{cdr}}\,({\it{l\/}}), {\it{i\/}} $-\;1$)\index{nth}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{l\/}} $\in\;${\bf{N}}${}$ \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\it{l\/}} \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\- \\ 
{\bf else }{\bf{f}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable nth.

\addvspace{18pt}

\begin{verbatim};;;update ith entry of l to be k
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{move} \\  
\=\+{\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{move} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{i\/}} $=$ {\tt{0}}$\;\;${\bf then }{\it{l\/}} \\ 
{\bf elseif }{\it{l\/}} $\simeq\;${{\bf{nil}}}${}$ \\ 
{\bf then }\=\+{\bf if }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\it{k\/}} \\ 
{\bf else }{\it{l\/}}$\;${\bf  endif}\- \\ 
{\bf elseif }{\it{i\/}} $=$ {\tt{1}}$\;\;${\bf then }{\rm{cons}}\,({\it{k\/}}, {\rm{cdr}}\,({\it{l\/}})) \\ 
{\bf else }{\rm{cons}}\,({\rm{car}}\,({\it{l\/}}), {\rm{move}}\,({\rm{cdr}}\,({\it{l\/}}), {\it{i\/}} $-\;1$, {\it{k\/}})\index{move})$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable move.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{at}$\;\;$
{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{at} $=$ ({\rm{nth}}\,({\it{l\/}}, {\it{i\/}})\index{nth} $=$ {\it{k\/}})
\end{tabbing}

\noindent{\sc Event}:   Disable at.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{length} \\  
\=\+{\rm{length}}\,({\it{l\/}})\index{length} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\rm{listp}}\,({\it{l\/}})$\;\;${\bf then }$1\;+$ {\rm{length}}\,({\rm{cdr}}\,({\it{l\/}}))\index{length} \\ 
{\bf else }{\sc{zero}}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable length.

\addvspace{18pt}

\begin{verbatim};;;The nth entry in l is in the list i.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{union-at-n}$\;\;$
{\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{union-at-n} $=$ ({\rm{nth}}\,({\it{l\/}}, {\it{n\/}})\index{nth} $\in$ {\it{i\/}})
\end{tabbing}

\noindent{\sc Event}:   Disable union-at-n.

\addvspace{18pt}

\begin{verbatim};;;Any entry in l is in the list i.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{all-union} \\  
\=\+{\rm{all-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{all-union} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{union-at-n} $\wedge$ {\rm{all-union}}\,({\it{l\/}}, {\it{n\/}} $-\;1$, {\it{i\/}})\index{all-union}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable all-union.

\addvspace{18pt}

\begin{verbatim};;;There exists an entry in l which belongs to 
;;;the list i, moreover when exists, some such
;;;j is returned.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{exist-union} \\  
\=\+{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{union-at-n}$\;\;${\bf then }{\it{n\/}} \\ 
{\bf else }{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}} $-\;1$, {\it{i\/}})\index{exist-union}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable exist-union.

\addvspace{18pt}

\begin{verbatim};;;n is in the intersection of l8-12 and g34.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{intersect-8-12-3-4-at-n} \\  
\=\+{\rm{intersect-8-12-3-4-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{intersect-8-12-3-4-at-n} \\ 
$=$$\;\;\;\;$({\rm{union-at-n}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}


\noindent{\sc Event}:   Disable intersect-8-12-3-4-at-n.

\addvspace{18pt}

\begin{verbatim};;;There exists n in the intersection of l8-12 and g34.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{exist-intersect-8-12-3-4} \\  
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{intersect-8-12-3-4-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{intersect-8-12-3-4-at-n}$\;\;${\bf then }{\it{n\/}} \\ 
{\bf else }{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}} $-\;1$, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable exist-intersect-8-12-3-4.

\addvspace{18pt}

\begin{verbatim};*Flag invariant.

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{lg-1-at-n} \\  
\=\+{\rm{lg-1-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-1-at-n} \\ 
$=$$\;\;\;\;$(\=\+({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{0}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{0}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{1}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{0}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{2}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{0}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{3}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{1}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{4}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{1}})\index{at}))\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-1-at-n.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{lg-2-at-n} \\  
\=\+{\rm{lg-2-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-2-at-n} \\ 
$=$$\;\;\;\;$(\=\+({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{5}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{3}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{6}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{3}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{7}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{2}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{8}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{3}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{8}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{2}})\index{at}))\-\-
\end{tabbing}     

\noindent{\sc Event}:   Disable lg-2-at-n.

\addvspace{18pt}       

\begin{tabbing}{\sc Definition}:\index{lg-3-at-n} \\  
\=\+{\rm{lg-3-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-3-at-n} \\ 
$=$$\;\;\;\;$(\=\+({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{9}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{10}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{11}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}) \\ 
$\vee$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{n\/}}, {\tt{12}})\index{at} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{n\/}}, {\tt{4}})\index{at}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable lg-3-at-n.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{lg-at-n} \\  
\=\+{\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n} \\ 
$=$$\;\;\;\;$({\rm{lg-1-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-1-at-n} $\wedge$ {\rm{lg-2-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-2-at-n} $\wedge$ {\rm{lg-3-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-3-at-n})\-
\end{tabbing}

\noindent{\sc Event}:   Disable lg-at-n.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{lg} \\  
\=\+{\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{t}} \\ 
{\bf else }{\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n} $\wedge$ {\rm{lg}}\,({\it{n\/}} $-\;1$, {\it{l\/}}, {\it{g\/}})\index{lg}$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable lg.

\addvspace{18pt}


\begin{verbatim};*The set {1...n}.
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{nset} \\  
\=\+{\rm{nset}}\,({\it{n\/}})\index{nset} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{nil}} \\ 
{\bf else }{\rm{cons}}\,({\it{n\/}}, {\rm{nset}}\,({\it{n\/}} $-\;1$)\index{nset})$\;${\bf  endif}\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable nset.

\addvspace{18pt}

\begin{verbatim};;;n belongs to nset.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-in-nset \\ \index{n-in-nset}
({\it{n\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ ({\it{n\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{verbatim};;;Any element in nset is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  nset-number \\ \index{nset-number}
({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\rightarrow$ ({\it{k\/}} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;If a nonzero number plus one belongs to nset, 
;;;then so does the nonzero number itself.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  add1-nset \\ \index{add1-nset}
(({\it{k\/}} $\not\simeq {\tt{0}}$) $\wedge$ (($1\;+$ {\it{k\/}}) $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})) $\rightarrow$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}                 

\begin{verbatim};;;Any list has its length at least nonzero.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  list-ln \\ \index{list-ln}
{\rm{listp}}\,({\it{l\/}}) $\rightarrow$ ({\rm{length}}\,({\it{l\/}})\index{length} $\not=$ {\tt{0}})
\end{tabbing}

\begin{verbatim};;;(move l k i) is again a list if l is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-is-list \\ \index{move-is-list}
{\rm{listp}}\,({\it{l\/}}) $\rightarrow$ {\rm{listp}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move})
\end{tabbing}


\noindent{\sc Event}:   Enable length.

\addvspace{18pt}

\begin{verbatim};;;(move l k i) has i as its kth entry.
;;;(enable length) is critical to prove this lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-nth \\ \index{move-nth}
({\rm{listp}}\,({\it{l\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset})) $\rightarrow$ ({\rm{nth}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{k\/}})\index{nth} $=$ {\it{i\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  zero-not-member-nset \\ \index{zero-not-member-nset}
{\tt{0}} $\not\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}
\end{tabbing}

\begin{verbatim};;;Lists l and (move l k i) have the same length.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-unchange-length \\ \index{move-unchange-length}
\=\+({\rm{listp}}\,({\it{l\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{length}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move})\index{length} $=$ {\rm{length}}\,({\it{l\/}})\index{length})\-
\end{tabbing}

\begin{verbatim};;;Lists l and (move l k i) have the same entries 
;;;except kth one.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  move-unchange-other-than-nth \\ \index{move-unchange-other-than-nth}
\=\+({\rm{listp}}\,({\it{l\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) $\wedge$ ({\it{j\/}} $\not=$ {\it{k\/}})) \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\it{i\/}})\index{move}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{l\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  member-ex-union \\ \index{member-ex-union}
{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\rightarrow$ ({\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{verbatim};;;(exist-union l n i) is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  number-ex-union \\ \index{number-ex-union}
{\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\rightarrow$ ({\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\it{i\/}})\index{exist-union} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;(exist-intersect-8-12-3-4 n l g) belongs to nset.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  member-intersect \\ \index{member-intersect}
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})\-
\end{tabbing}

\begin{verbatim};;;(exist-intersect-8-12-3-4 n l g) is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  number-intersect \\ \index{number-intersect}
{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\rightarrow$ ({\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;any member of nset is nonzero.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-not-0 \\ \index{k-not-0}
({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\rightarrow$ ({\it{k\/}} $\not\simeq {\tt{0}}$)
\end{tabbing}

\begin{verbatim};*lemmas for a0 

;;;If j's entry in l is between 8..12 then  
;;;(exist-union l n '(8 9 10 11 12)) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-ex-l8-12 \\ \index{j-ex-l8-12}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}\-
\end{tabbing}

\begin{verbatim};;;Witness of (exist-union lp n '(8 9 10 11 12)) 
;;;has in lp its entry between 8...12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-lp8-12-in-lp8-12 \\ \index{ex-lp8-12-in-lp8-12}
\=\+{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,(\=\+{\it{lp\/}}, \\ 
{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, \\ 
{\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\-\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union l n '(8 9 10 11 12))) 
;;;holds, then (not (exist-union g n '(4))) by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-if4 \\ \index{ex-if4}
\=\+(($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{4}}{\tt{)}})\index{exist-union})\-
\end{tabbing}

\begin{verbatim};;;If (not (exist-union g n '(1))) holds,
;;; then there is no entry either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l34-empty \\ \index{l34-empty}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ ($\neg$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{1}}{\tt{)}})\index{exist-union})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If j's entry in lp is 4, then (certainly)
;;;it is either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp4-then-un34 \\ \index{lp4-then-un34}
{\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{4}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If (exist-intersect-8-12-3-4 n l g) holds, 
;;;then so does (exist-union g n '(3 4)).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  int-8-12-3-4-then-un34 \\ \index{int-8-12-3-4-then-un34}
{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} $\rightarrow$ {\rm{exist-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{exist-union}
\end{tabbing}

\begin{verbatim};*lemmas for a1

;;;i is the witness of 
;;;(exist-intersect-8-12-3-4 n lp gp).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  int-wtn \\ \index{int-wtn}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{intersect-8-12-3-4-at-n}}\,({\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{exist-intersect-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;If there exists j such that j's entry in lp 
;;;is between 8..12 and entry in gp is either 3 or 4
;;;then (intersect-8-12-3-4-at-n j lp gp) holds.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-12-and-un34-then-int \\ \index{un8-12-and-un34-then-int}
\=\+({\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{j\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{verbatim};;;By the two lemmas above, 
;;;(exist-intersect-8-12-3-4 n lp gp) holds provided 
;;;that there exists j such that j's entry in lp is
;;;between 8..12 and entry in gp is either 3 or 4.

;* ep-l8-12

;;;If the k's entry in l is 5, then the k's entry 
;;;in g is 3 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lg-l5-g3 \\ \index{lg-l5-g3}
(({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}) $\rightarrow$ {\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{at}
\end{tabbing}

\begin{verbatim};;;If the k's entry in gp is 3 then certainly 
;;;it is either 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  gp3-then-un34 \\ \index{gp3-then-un34}
{\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{3}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;nep-l8-12

;;;If the k's entry in l is between 8..12 then 
;;;it is either between 8..11 or equal to 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  case-k \\ \index{case-k}
\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;;;k-not-l8-12

;;;If (exist-intersect-8-12-3-4 n l g) holds
;;;then the witness has its entry in g either equal 
;;;to 3 or 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  intersect-8-12-3-4-then-3-4 \\ \index{intersect-8-12-3-4-then-3-4}
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If (exist-intersect-8-12-3-4 n l g) holds,
;;; then the witness has its entry in g between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  intersect-8-12-3-4-then-8-12 \\ \index{intersect-8-12-3-4-then-8-12}
\=\+{\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\rm{exist-intersect-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{exist-intersect-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;k-in-l8-11

;;;If k's entry in lp is between 9 and 12,
;;;then it is certainly between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un9-12-then-un8-12 \\ \index{un9-12-then-un8-12}
\=\+{\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the i's entry in l is between 9 and 12,
;;;then the k's entry in g is 4.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  if4 \\ \index{if4}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;k-in-l12

;;;If (exist-union lp n '(8 9 10 11 12)) holds then
;;;its witness does not have its entry in lp equal to 1.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-lp8-12-not-in-lp0 \\ \index{ex-lp8-12-not-in-lp0}
\=\+{\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union} \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\rm{exist-union}}\,({\it{lp\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union}, {\tt{0}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;If k's entry in lp is between 8 and 12,
;;; then it is either between 8 and 11 or 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp9-12-or-lp8 \\ \index{k-in-lp9-12-or-lp8}
\=\+(\=\+{\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry is either 5 or 7,
;;;then it is between 5 and 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un57-then-un5-12 \\ \index{un57-then-un5-12}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 8 and 11,
;;;then it is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-11-then-un5-12 \\ \index{un8-11-then-un5-12}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 8 and 12,
;;;then it is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-12-then-un5-12 \\ \index{un8-12-then-un5-12}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}


\begin{verbatim};*lemmas for a2

;;;i-eq-k-j-neq-k

;;;If the k's entry in l is either 10 or 11, 
;;;then the k's entry in l is between 10 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un10-11-then-un10-12 \\ \index{un10-11-then-un10-12}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the j's entry in g is either 0 or 1 then
;;;the j's entry in l is not between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  if1 \\ \index{if1}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;j-eq-k-i-neq-k

;;;If the k's entry in l is between 5 and 7,
;;;then it is certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un5-7-then-un5-11 \\ \index{un5-7-then-un5-11}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 5 and 7 then 
;;;it is certain between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un57-then-un5-11 \\ \index{un57-then-un5-11}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 8 and 11,
;;;then it is certainly between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-11-then-un5-11 \\ \index{un8-11-then-un5-11}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 5 and 12 and
;;;the k's entry in lp is between 5 and 7, then
;;;the k's entry in lp in fact is between 9 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-lp5-7-or-lp8-or-lp9-12 \\ \index{k-in-lp5-7-or-lp8-or-lp9-12}
\=\+(\=\+{\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 5 and 11,
;;; then it is certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un5-11-then-un5-12 \\ \index{un5-11-then-un5-12}
\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 10 and 12,
;;; then it is certainly between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un10-12-then-un8-12 \\ \index{un10-12-then-un8-12}
{\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;j-eq-k-i-neq-k

;;;If (exist-union l n '(8 9 10 11 12)) does not hold,
;;;then the i's entry in l is not between 10 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-not-l10-12 \\ \index{i-not-l10-12}
\=\+(({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}


\begin{verbatim};*lemmas for a3

;;;j-eq-k-i-neq-k

;;;If the k's entry in l is between 5 and 11,
;;;then the k's entry in l is between 9 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un5-11-eq-un58-or-un8-11 \\ \index{un5-11-eq-un58-or-un8-11}
\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in g is 4,
;;;then the k's entry in l is between 5 and 8.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  a3-if4 \\ \index{a3-if4}
\=\+(({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is between 5 and 11,
;;;and the k's entry in l is between 5 and 12,
;;;then the k's entry in l is 9 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-in-l5-11-g4-then-l9-11 \\ \index{k-in-l5-11-g4-then-l9-11}
\=\+(\=\+({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the i's entry in l is 12, 
;;;then the i's entry in l is between 8 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l12-then-un8-12 \\ \index{l12-then-un8-12}
{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If (exist-union l n '(8 9 10 11 12)) does not hold,
;;;then the i's entry in l is 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  i-not-in-l12 \\ \index{i-not-in-l12}
\=\+(({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ($\neg$ {\rm{exist-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{exist-union})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;j-neq-k-i-eq-k

;;;If the k's entry in l is 11,
;;; then the k's entry in l is between 10 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l11-then-un10-12 \\ \index{l11-then-un10-12}
{\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};;;If the j's entry in g is either 2 or 3,
;;;then the j's entry in l is between 5 and 8 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  if3 \\ \index{if3}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ ($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}


\begin{verbatim};;;If the j's entry in l is between 5 and 12 and
;;;the j's entry in l is between 5 and 8, then
;;;the j's entry in l is 9 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l5-12-eq-l5-8-or-l9-12 \\ \index{l5-12-eq-l5-8-or-l9-12}
\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;i-j-eq-k

;;;If the k's entry in lp is 12,
;;;then it is certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l12-then-un9-12 \\ \index{l12-then-un9-12}
{\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{verbatim};*lemmas for b1a

;;;If the u's entry in g is 4,
;;;then the u's entry in l is between 8 and 12 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b1a-if4 \\ \index{b1a-if4}
\=\+(({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{u\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};*lemmas for b1b

;;;If the k's entry in lp is between 9 and 12,
;;;then the k's entry in gp is iether 3 or 4 by lg.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp9-12-then-k-in-g34 \\ \index{lp9-12-then-k-in-g34}
\=\+(({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 8 and 12, and
;;;it is not 8, then it is certainly between 9 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-12-then-l8-or-l9-12 \\ \index{un8-12-then-l8-or-l9-12}
\=\+({\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ ($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at})) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}




\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; moldefn.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;* Well-formed states

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{molws} \\  
\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$=$$\;\;\;\;$(\=\+({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{h\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{h\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{l\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{ }}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{g\/}}, {\it{n\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{all-union} \\ 
$\wedge$$\;\;\;\;${\rm{all-union}}\,({\it{h\/}}, {\it{n\/}}, {\rm{nset}}\,($1\;+$ {\it{n\/}})\index{nset})\index{all-union})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable molws.

\addvspace{18pt}

\begin{verbatim};* Transitions

\end{verbatim}\begin{tabbing}{\sc Definition}:\index{mrhoi0} \\  
\=\+{\rm{mrhoi0}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi0} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{0}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{1}})\index{move}) $\wedge$ ({\it{hp\/}} $=$ {\it{h\/}}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi1a} \\  
\=\+{\rm{mrhoi1a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi1a} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{1}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ ({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{2}})\index{move}) $\wedge$ ({\it{hp\/}} $=$ {\it{h\/}}))\-
\end{tabbing}
\begin{tabbing}{\sc Definition}:\index{mrhoi1b} \\  
\=\+{\rm{mrhoi1b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi1b} \\ 
$=$$\;\;\;\;$({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{1}})\index{at} $\wedge$ ({\it{gp\/}} $=$ {\it{g\/}}) $\wedge$ ({\it{lp\/}} $=$ {\it{l\/}}) $\wedge$ ({\it{hp\/}} $=$ {\it{h\/}}))\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi2} \\  
\=\+{\rm{mrhoi2}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi2} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{3}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{1}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, {\tt{1}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi3a} \\  
\=\+{\rm{mrhoi3a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi3a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\it{h\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\it{n\/}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{4}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi3b} \\  
\=\+{\rm{mrhoi3b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi3b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $<$ ($1\;+$ {\it{n\/}})) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth})\index{move}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{)}})\index{union-at-n})\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi4} \\  
\=\+{\rm{mrhoi4}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi4} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{3}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, {\tt{1}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi5a} \\  
\=\+{\rm{mrhoi5a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\it{h\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\it{n\/}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{8}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi5b} \\  
\=\+{\rm{mrhoi5b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\it{h\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $<$ ($1\;+$ {\it{n\/}})) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{6}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi5c} \\  
\=\+{\rm{mrhoi5c}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5c} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $<$ ($1\;+$ {\it{n\/}})) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{1}})\index{at}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi6} \\  
\=\+{\rm{mrhoi6}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi6} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{2}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{7}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, {\tt{1}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi7a} \\  
\=\+{\rm{mrhoi7a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi7a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{8}})\index{move}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\it{h\/}}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi7b} \\  
\=\+{\rm{mrhoi7b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi7b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ (({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $-\;1$) ${}${\bf{mod}}${}$ {\it{n\/}}))\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi8} \\  
\=\+{\rm{mrhoi8}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi8} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{4}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, {\tt{1}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi9a} \\  
\=\+{\rm{mrhoi9a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi9a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{i\/}}, {\it{i\/}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{10}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\it{h\/}}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi9b} \\  
\=\+{\rm{mrhoi9b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi9b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\it{l\/}}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi10} \\  
\=\+{\rm{mrhoi10}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi10} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\it{i\/}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi11a} \\  
\=\+{\rm{mrhoi11a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi11a} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\it{n\/}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\it{h\/}}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi11b} \\  
\=\+{\rm{mrhoi11b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi11b} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $<$ ($1\;+$ {\it{n\/}})) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\rm{move}}\,({\it{h\/}}, {\it{i\/}}, $1\;+$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\it{l\/}}))\-\-
\end{tabbing}


\begin{tabbing}{\sc Definition}:\index{mrhoi12} \\  
\=\+{\rm{mrhoi12}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi12} \\ 
$=$$\;\;\;\;$(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{hp\/}} $=$ {\it{h\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{gp\/}} $=$ {\rm{move}}\,({\it{g\/}}, {\it{i\/}}, {\tt{0}})\index{move}) \\ 
$\wedge$$\;\;\;\;$({\it{lp\/}} $=$ {\rm{move}}\,({\it{l\/}}, {\it{i\/}}, {\tt{0}})\index{move}))\-\-
\end{tabbing}

\begin{tabbing}{\sc Definition}:\index{mrhoi} \\  
\=\+{\rm{mrhoi}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$=$$\;\;\;\;$(\=\+{\rm{mrhoi0}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi0} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi1a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi1a} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi1b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi1b} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi2}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi2} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi3a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi3a} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi3b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi3b} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi4}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi4} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi5a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5a} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi5b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5b} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi5c}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5c} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi6}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi6} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi7a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi7a} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi7b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi7b} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi8}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi8} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi9a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi9a} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi9b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi9b} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi10}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi10} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi11a}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi11a} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi11b}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi11b} \\ 
$\vee$$\;\;\;\;${\rm{mrhoi12}}\,({\it{n\/}}, {\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi12})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable mrhoi.

\addvspace{18pt}

\begin{verbatim};*  Invariants

;;;b0
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{b0a} \\  
{\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a} $=$ (({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} $\wedge$ ({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth})) $\rightarrow$ ($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at}))
\end{tabbing}

\noindent{\sc Event}:   Disable b0a.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{b0b} \\  
\=\+{\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b} \\ 
$=$$\;\;\;\;$(({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} $\wedge$ ({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}) $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at}) $\rightarrow$ ({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth}))\-
\end{tabbing}

\noindent{\sc Event}:   Disable b0b.

\addvspace{18pt}

\begin{verbatim};;;b1
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{b1a} \\  
{\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} $=$ ({\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\rightarrow$ ($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at}))
\end{tabbing}

\noindent{\sc Event}:   Disable b1a.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{hint-8-12-3-4-at-n} \\  
\=\+{\rm{hint-8-12-3-4-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{hint-8-12-3-4-at-n} \\ 
$=$$\;\;\;\;$({\rm{intersect-8-12-3-4-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{intersect-8-12-3-4-at-n} $\wedge$ ({\it{n\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth}))\-
\end{tabbing}


\noindent{\sc Event}:   Disable hint-8-12-3-4-at-n.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{exist-hint-8-12-3-4} \\  
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$=$$\;\;\;\;$\=\+{\bf if }{\it{n\/}} $\simeq {\tt{0}}$$\;\;${\bf then }{\bf{f}} \\ 
{\bf elseif }{\rm{hint-8-12-3-4-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{hint-8-12-3-4-at-n}$\;\;${\bf then }{\it{n\/}} \\ 
{\bf else }{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}} $-\;1$, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}$\;${\bf  endif}\-\-
\end{tabbing}


\noindent{\sc Event}:   Disable exist-hint-8-12-3-4.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{b1b} \\  
\=\+{\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b} \\ 
$=$$\;\;\;\;$(\=\+({\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable b1b.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{b1c} \\  
\=\+{\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$=$$\;\;\;\;$(\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{4}})\index{at}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable b1c.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{b1d} \\  
{\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}})\index{b1d} $=$ ({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{7}})\index{at} $\rightarrow$ ({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}))
\end{tabbing}

\noindent{\sc Event}:   Disable b1d.

\addvspace{18pt}

\begin{verbatim};;;; b2
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{b2a} \\  
\=\+{\rm{b2a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a} \\ 
$=$$\;\;\;\;$(\=\+(({\it{j\/}} $<$ {\it{i\/}}) $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable b2a.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{b2b} \\  
\=\+{\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b} \\ 
$=$$\;\;\;\;$(\=\+(({\it{j\/}} $<$ {\it{i\/}}) $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} $\wedge$ ({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable b2b.

\addvspace{18pt}

\begin{verbatim};;;b3
\end{verbatim}\begin{tabbing}{\sc Definition}:\index{b3a} \\  
\=\+{\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{j\/}})\index{b3a} \\ 
$=$$\;\;\;\;$(\=\+({\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable b3a.

\addvspace{18pt}

\begin{tabbing}{\sc Definition}:\index{b3b} \\  
\=\+{\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b} \\ 
$=$$\;\;\;\;$(\=\+(\=\+{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-\-
\end{tabbing}

\noindent{\sc Event}:   Disable b3b.

\addvspace{18pt}



\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; molbasic.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
\end{verbatim}\begin{tabbing}{\sc Theorem}:  hint-member \\ \index{hint-member}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  n-not-less-j \\ \index{n-not-less-j}
({\it{n\/}} $<$ {\it{j\/}}) $\rightarrow$ ({\it{j\/}} $\not\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{verbatim};;;molws implies that n is a number.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-num-n \\ \index{molws-num-n}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ ({\it{n\/}} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;molws implies that l is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-list-l \\ \index{molws-list-l}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ {\rm{listp}}\,({\it{l\/}})
\end{tabbing}

\begin{verbatim};;;molws implies that g is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-list-g \\ \index{molws-list-g}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ {\rm{listp}}\,({\it{g\/}})
\end{tabbing}

\begin{verbatim};;;molws implies that h is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-list-h \\ \index{molws-list-h}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ {\rm{listp}}\,({\it{h\/}})
\end{tabbing}

\begin{verbatim};;;molws implies that length of l is n.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-ln-l \\ \index{molws-ln-l}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ ({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}})
\end{tabbing}

\begin{verbatim};;;molws implies that length of g is n.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-ln-g \\ \index{molws-ln-g}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ ({\rm{length}}\,({\it{g\/}})\index{length} $=$ {\it{n\/}})
\end{tabbing}

\begin{verbatim};;;molws implies that length of h is n.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-ln-h \\ \index{molws-ln-h}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ ({\rm{length}}\,({\it{h\/}})\index{length} $=$ {\it{n\/}})
\end{tabbing}

\begin{verbatim};;;molws and mrho imply that lp is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-ln-lp \\ \index{molws-ln-lp}
\=\+({\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi}) \\ 
$\rightarrow$$\;\;\;\;${\rm{listp}}\,({\it{lp\/}})\-
\end{tabbing}

\begin{verbatim};;;molws and mrho imply that gp is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-ln-gp \\ \index{molws-ln-gp}
\=\+({\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi}) \\ 
$\rightarrow$$\;\;\;\;${\rm{listp}}\,({\it{gp\/}})\-
\end{tabbing}

\begin{verbatim};;;molws and mrho imply that hp is a list.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-ln-hp \\ \index{molws-ln-hp}
\=\+({\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi}) \\ 
$\rightarrow$$\;\;\;\;${\rm{listp}}\,({\it{hp\/}})\-
\end{tabbing}

\begin{verbatim};;;Another version of nset-number. 
;;;This is available in the theorem 
;;;where molws is disabled.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-num-k \\ \index{molws-num-k}
({\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})) $\rightarrow$ ({\it{k\/}} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  molws-union-h \\ \index{molws-union-h}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ {\rm{all-union}}\,({\it{h\/}}, {\it{n\/}}, {\rm{nset}}\,($1\;+$ {\it{n\/}})\index{nset})\index{all-union}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-nth-numberp \\ \index{lm-nth-numberp}
(({\it{i\/}} $\in\;${\bf{N}}${}$) $\wedge$ {\rm{all-union}}\,({\it{h\/}}, {\it{n\/}}, {\rm{nset}}\,({\it{i\/}})\index{nset})\index{all-union} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})) $\rightarrow$ ({\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nth-numberp \\ \index{nth-numberp}
({\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})) $\rightarrow$ ({\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth} $\in\;${\bf{N}}${}$)
\end{tabbing}

\begin{verbatim};;;molws implies that n is nonzero.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  molws-n-not-0 \\ \index{molws-n-not-0}
{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\rightarrow$ ({\it{n\/}} $\not=$ {\tt{0}})
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-l-mrholemma \\ \index{lm-l-mrholemma}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{l\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{lp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}


\noindent{\sc Event}:   Disable lm-l-mrholemma.

\addvspace{18pt}

\begin{verbatim};;;Mrholemma for list l.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l-mrholemma \\ \index{l-mrholemma}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{l\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{lp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-g-mrholemma \\ \index{lm-g-mrholemma}
\=\+(\=\+{\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{g\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{g\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{g\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{gp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}
 

\noindent{\sc Event}:   Disable lm-g-mrholemma.

\addvspace{18pt}

\begin{verbatim};;;Mrholemma for list g.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  g-mrholemma \\ \index{g-mrholemma}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{g\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{gp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{verbatim};;;Auxiliary lemma.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-h-mrholemma \\ \index{lm-h-mrholemma}
\=\+(\=\+{\rm{listp}}\,({\it{h\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{h\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{h\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{hp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}
 

\noindent{\sc Event}:   Disable lm-h-mrholemma.

\addvspace{18pt}

\begin{verbatim};;;Mrholemma for list g.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  h-mrholemma \\ \index{h-mrholemma}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth} $=$ {\rm{nth}}\,({\it{hp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{verbatim};;; lp-gp-same-l-g

;;;Another version of Rholemma for l.
;;;It applies to (union-at-n l j m) in stead of 
;;;(nth l j).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-lp-same-l \\ \index{m-lp-same-l}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;Contrast to the one above, 
;;;the order of l and lp is reversed.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-l-same-lp \\ \index{m-l-same-lp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-lp-same-l-not \\ \index{m-lp-same-l-not}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;Another version of Rholemma for g.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-gp-same-g \\ \index{m-gp-same-g}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;Contrast to the one above, 
;;;the order of g and gp is reversed.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-g-same-gp \\ \index{m-g-same-gp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-gp-same-g-not \\ \index{m-gp-same-g-not}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;Another version of Rholemma for h.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-hp-same-h \\ \index{m-hp-same-h}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{h\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{hp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;Contrast to the one above, 
;;;the order of g and gp is reversed.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-h-same-hp \\ \index{m-h-same-hp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{m\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{hp\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{h\/}}, {\it{j\/}}, {\it{m\/}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;It applies to (at l j m) in stead of 
;;;(nth l j).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-l-same-lp-at \\ \index{m-l-same-lp-at}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{m\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-gp-same-g-at \\ \index{m-gp-same-g-at}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{m\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{j\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\it{m\/}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{j\/}}, {\it{m\/}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-l-same-lp-at-not \\ \index{m-l-same-lp-at-not}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{m\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\it{m\/}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\it{m\/}})\index{at})\-
\end{tabbing}



\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; mollg.ev ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;mrhoi0
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi0 \\ \index{n-neq-k-mrhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi0 \\ \index{n-eq-k-mrhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi0 \\ \index{lg-at-mrhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi0 \\ \index{lg-mrhoi0}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi0.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi0-preserves-lg \\ \index{mrhoi0-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi0}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi0} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi1a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi1a \\ \index{n-neq-k-mrhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi1a \\ \index{n-eq-k-mrhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi1a \\ \index{lg-at-mrhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi1a \\ \index{lg-mrhoi1a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{1}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi1a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi1a-preserves-lg \\ \index{mrhoi1a-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi1a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi1a} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi1b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  mrhoi1b-preserves-lg \\ \index{mrhoi1b-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi1b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi1b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi2
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi2 \\ \index{n-neq-k-mrhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi2 \\ \index{n-eq-k-mrhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi2 \\ \index{lg-at-mrhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi2 \\ \index{lg-mrhoi2}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{1}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi2.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi2-preserves-lg \\ \index{mrhoi2-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi2}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi2} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi3a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi3a \\ \index{n-neq-k-mrhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi3a \\ \index{n-eq-k-mrhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi3a \\ \index{lg-at-mrhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi3a \\ \index{lg-mrhoi3a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi3a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi3a-preserves-lg \\ \index{mrhoi3a-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi3a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi3a} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi3b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  mrhoi3b-preserves-lg \\ \index{mrhoi3b-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi3b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi3b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi4
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi4 \\ \index{n-neq-k-mrhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi4 \\ \index{n-eq-k-mrhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi4 \\ \index{lg-at-mrhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi4 \\ \index{lg-mrhoi4}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{3}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi4.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi4-preserves-lg \\ \index{mrhoi4-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi4}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi4} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi5a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi5a \\ \index{n-neq-k-mrhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi5a \\ \index{n-eq-k-mrhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi5a \\ \index{lg-at-mrhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi5a \\ \index{lg-mrhoi5a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi5a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi5a-preserves-lg \\ \index{mrhoi5a-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi5a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5a} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi5b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi5b \\ \index{n-neq-k-mrhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi5b \\ \index{n-eq-k-mrhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi5b \\ \index{lg-at-mrhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi5b \\ \index{lg-mrhoi5b}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi5b.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi5b-preserves-lg \\ \index{mrhoi5b-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi5b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi5c
\end{verbatim}\begin{tabbing}{\sc Theorem}:  mrhoi5c-preserves-lg \\ \index{mrhoi5c-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi5c}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi5c} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi6
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi6 \\ \index{n-neq-k-mrhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi6 \\ \index{n-eq-k-mrhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi6 \\ \index{lg-at-mrhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi6 \\ \index{lg-mrhoi6}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{6}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{2}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi6.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi6-preserves-lg \\ \index{mrhoi6-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi6}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi6} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi7a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi7a \\ \index{n-neq-k-mrhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi7a \\ \index{n-eq-k-mrhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi7a \\ \index{lg-at-mrhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi7a \\ \index{lg-mrhoi7a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi7a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi7a-preserves-lg \\ \index{mrhoi7a-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi7a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi7a} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi7b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  mrhoi7b-preserves-lg \\ \index{mrhoi7b-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi7b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi7b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi8
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi8 \\ \index{n-neq-k-mrhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi8 \\ \index{n-eq-k-mrhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi8 \\ \index{lg-at-mrhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi8 \\ \index{lg-mrhoi8}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi8.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi8-preserves-lg \\ \index{mrhoi8-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi8}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi8} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi9a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi9a \\ \index{n-neq-k-mrhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi9a \\ \index{n-eq-k-mrhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi9a \\ \index{lg-at-mrhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi9a \\ \index{lg-mrhoi9a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi9a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi9a-preserves-lg \\ \index{mrhoi9a-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi9a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi9a} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi9b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  mrhoi9b-preserves-lg \\ \index{mrhoi9b-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi9b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi9b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi10
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi10 \\ \index{n-neq-k-mrhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi10 \\ \index{n-eq-k-mrhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi10 \\ \index{lg-at-mrhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi10 \\ \index{lg-mrhoi10}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi10.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi10-preserves-lg \\ \index{mrhoi10-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi10}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi10} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi11a
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi11a \\ \index{n-neq-k-mrhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-neq-k-mrhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi11a \\ \index{n-eq-k-mrhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi11a \\ \index{lg-at-mrhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi11a \\ \index{lg-mrhoi11a}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{move}, {\it{g\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi11a.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi11a-preserves-lg \\ \index{mrhoi11a-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi11a}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi11a} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi11b
\end{verbatim}\begin{tabbing}{\sc Theorem}:  mrhoi11b-preserves-lg \\ \index{mrhoi11b-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi11b}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi11b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{verbatim};;;mrhoi12
\end{verbatim}\begin{tabbing}{\sc Theorem}:  n-neq-k-mrhoi12 \\ \index{n-neq-k-mrhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg-at-n}\-
\end{tabbing}
 

\noindent{\sc Event}:   Disable n-neq-k-mrhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  n-eq-k-mrhoi12 \\ \index{n-eq-k-mrhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{k\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable n-eq-k-mrhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-at-mrhoi12 \\ \index{lg-at-mrhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg-at-n}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-at-mrhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lg-mrhoi12 \\ \index{lg-mrhoi12}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{g\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\rm{length}}\,({\it{l\/}})\index{length})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\rm{move}}\,({\it{l\/}}, {\it{k\/}}, {\tt{0}})\index{move}, {\rm{move}}\,({\it{g\/}}, {\it{k\/}}, {\tt{0}})\index{move})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lg-mrhoi12.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  mrhoi12-preserves-lg \\ \index{mrhoi12-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi12}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi12} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrho-preserves-lg \\ \index{mrho-preserves-lg}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg}\-
\end{tabbing}


\noindent{\sc Event}:   Disable mrhoi0-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi1a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi1b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi2-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi3a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi3b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi4-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi5a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi5b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi5c-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi6-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi7a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi7b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi8-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi9a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi9b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi10-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi11a-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi11b-preserves-lg.

\addvspace{18pt}

\noindent{\sc Event}:   Disable mrhoi12-preserves-lg.

\addvspace{18pt}



\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;   b0.ev   ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;   b0a   ;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;Common in mole and atom.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b0a-if1 \\ \index{b0a-if1}
(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ ($\neg$ {\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{1}})\index{at})) $\rightarrow$ ($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})
\end{tabbing}
\begin{verbatim};;;;;;;;;;;;;comm end.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  if1-nth-h-k \\ \index{if1-nth-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{1}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l5-not-g1 \\ \index{l5-not-g1}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{1}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l5-nth-h-k-eq-j \\ \index{l5-nth-h-k-eq-j}
\=\+(\=\+{\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l5-j-lt-nth-k \\ \index{l5-j-lt-nth-k}
\=\+(\=\+({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  nth-k-lt-j-or-eq-j \\ \index{nth-k-lt-j-or-eq-j}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-j-not-in-l4 \\ \index{lm-j-not-in-l4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cond-l5 \\ \index{cond-l5}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-not-in-l4 \\ \index{j-not-in-l4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l5 \\ \index{k-in-l5}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;The order of the hints is crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b0a-i-eq-k-j-neq-k \\ \index{lm-b0a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0a-i-eq-k-j-neq-k \\ \index{b0a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0a-i-j-eq-k \\ \index{b0a-i-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{k\/}})\index{b0a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{b0a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0a-i-eq-k \\ \index{b0a-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a}\-
\end{tabbing}

\begin{verbatim};;;n-not-less-j is necessary.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  cond-lp4 \\ \index{cond-lp4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  not-l3-then-lp4 \\ \index{not-l3-then-lp4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-in-l5 \\ \index{i-in-l5}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b0a-i-neq-k-j-eq-k \\ \index{lm-b0a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0a-i-neq-k-j-eq-k \\ \index{b0a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{k\/}})\index{b0a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0a-i-j-neq-k \\ \index{b0a-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0a-i-neq-k \\ \index{b0a-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-b0a \\ \index{rho-preserves-b0a}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a}\-
\end{tabbing}


\begin{verbatim};;;;;;;;;;;;;   b0b   ;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;;;;Common in mole and atom.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b0b-if1 \\ \index{b0b-if1}
(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at}) $\rightarrow$ {\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{1}})\index{at}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-if3 \\ \index{b0b-if3}
\=\+(({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}
\begin{verbatim};;;;;;;;;;;;;;;;;Common in mole and atom end.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-j-neq-h-k \\ \index{lm-j-neq-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{1}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-not-g1 \\ \index{h-k-not-g1}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{1}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-neq-h-k \\ \index{j-neq-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  n-k-leq-sub1-i \\ \index{n-k-leq-sub1-i}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{i\/}})\index{at}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\it{i\/}} $-\;1$) $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{verbatim};;;This is proved with help of n-k-leq-sub1-i.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm1-j-in-l3 \\ \index{lm1-j-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-j-in-l3 \\ \index{lm-j-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{verbatim};;;The order of hints is crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-in-l3 \\ \index{j-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b0b-i-eq-k-j-neq-k \\ \index{lm-b0b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-i-eq-k-j-neq-k \\ \index{b0b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-i-j-eq-k \\ \index{b0b-i-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{b0b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-i-eq-k \\ \index{b0b-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-i-neq-h-k \\ \index{lm-i-neq-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{i\/}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-g02 \\ \index{h-k-g02}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-neq-h-k \\ \index{i-neq-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{i\/}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm1-k-in-l3-imp \\ \index{lm1-k-in-l3-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\it{i\/}} $-\;1$) $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-k-in-l3-imp \\ \index{lm-k-in-l3-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\it{i\/}} $-\;1$) $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cond-l3 \\ \index{cond-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\it{i\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l3-imp \\ \index{k-in-l3-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l2-imp \\ \index{k-in-l2-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lp3-then-l2-or-l3 \\ \index{lp3-then-l2-or-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-i-in-l5 \\ \index{b0b-i-in-l5}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b0b-i-neq-k-j-eq-k \\ \index{lm-b0b-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{i\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-i-neq-k-j-eq-k \\ \index{b0b-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{k\/}})\index{b0b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-i-j-neq-k \\ \index{b0b-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b0b-i-neq-k \\ \index{b0b-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  rho-preserves-b0b \\ \index{rho-preserves-b0b}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b}\-
\end{tabbing}



\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;   b1.ev   ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;   b1a   ;;;;;;;;;;;;;;;;

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-h-k-eq-add1-n-nex-hint \\ \index{lm-h-k-eq-add1-n-nex-hint}
\=\+(({\it{n\/}} $\not\simeq {\tt{0}}$) $\wedge$ {\rm{listp}}\,({\it{h\/}}) $\wedge$ ({\it{n\/}} $\not<$ {\it{i\/}}) $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ ({\it{n\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{exist-hint-8-12-3-4}}\,({\it{i\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-eq-add1-n-nex-hint \\ \index{h-k-eq-add1-n-nex-hint}
\=\+({\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} $\wedge$ ({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{h\/}}, {\it{k\/}}, $1\;+$ {\it{n\/}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-eq-add1-n-k-not-in-l3 \\ \index{h-k-eq-add1-n-k-not-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, $1\;+$ {\it{n\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  not-l3-then-not-lp4 \\ \index{not-l3-then-not-lp4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-eq-add1-n \\ \index{h-k-eq-add1-n}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, $1\;+$ {\it{n\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-neq-add1-n \\ \index{h-k-neq-add1-n}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{k\/}}, $1\;+$ {\it{n\/}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;The order of the hints is crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b1a-i-neq-k-j-eq-k \\ \index{lm-b1a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;need m-l-same-lp.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b1a-i-neq-k-j-eq-k \\ \index{b1a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1a-i-j-neq-k \\ \index{b1a-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1a-i-neq-k \\ \index{b1a-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cond-l7 \\ \index{cond-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l7-imp \\ \index{k-in-l7-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l5-j-lt-h-k \\ \index{l5-j-lt-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l5-then-j-not-l4 \\ \index{k-in-l5-then-j-not-l4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lp9-12-k-in-l8-12 \\ \index{lp9-12-k-in-l8-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-lp9-12-then-j-not-l4 \\ \index{k-in-lp9-12-then-j-not-l4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-not-in-l7-then-lp9-12-or-l5 \\ \index{k-not-in-l7-then-lp9-12-or-l5}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-not-l7-imp \\ \index{k-in-not-l7-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;I wonder why the following two do not imply 
;;;lm-b1a-i-eq-k-j-neq-k although those without
;;;(member u (nset n)) are perfectly able to
;;;imply it.
;;;(prove-lemma k-in-lp9-12-then-j-not-l4 (rewrite)
;;;   (implies (and (molws n l g h)
;;;                 (member j (nset n))
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (b1a l k j)
;;;                 (union-at-n lp k '(9 10 11 12)))
;;;            (not (at l j 4))))
;;;
;;;(prove-lemma k-in-lp8-then-j-not-l4 (rewrite)
;;;   (implies (and (at lp k 8)
;;;                 (molws n l g h)
;;;                 (member j (nset n))
;;;                 (member u (nset n))
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (lg n l g)
;;;                 (b0a n l h k j)
;;;                 (b1a l (nth h k) j))
;;;            (not (at l j 4))))

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b1a-i-eq-k-j-neq-k \\ \index{lm-b1a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;m-l-same-lp-at-not is used.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b1a-i-eq-k-j-neq-k \\ \index{b1a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1a}}\,({\it{lp\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1a-i-j-eq-k \\ \index{b1a-i-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{k\/}}, {\it{k\/}})\index{b1a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1a}}\,({\it{lp\/}}, {\it{k\/}}, {\it{k\/}})\index{b1a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1a-i-eq-k \\ \index{b1a-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1a}}\,({\it{lp\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrho-preserves-b1a \\ \index{mrho-preserves-b1a}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0a}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a}\-
\end{tabbing}


\begin{verbatim};;;;;;;;;;;;;   b1b   ;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;;;;common in mole and atom.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  un8-11-then-un8-12 \\ \index{un8-11-then-un8-12}
{\rm{union-at-n}}\,({\it{lp\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} $\rightarrow$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l8-11-k-in-gp34 \\ \index{l8-11-k-in-gp34}
\=\+(({\it{r\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  u-if4 \\ \index{u-if4}
(({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at}) $\rightarrow$ ($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{u\/}}, {\tt{2}})\index{at})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l12-then-un10-12 \\ \index{l12-then-un10-12}
{\rm{at}}\,({\it{l\/}}, {\it{u\/}}, {\tt{12}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{u\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  r-neq-k \\ \index{r-neq-k}
({\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} $\wedge$ {\rm{at}}\,({\it{l\/}}, {\it{r\/}}, {\tt{12}})\index{at}) $\rightarrow$ ({\it{k\/}} $\not=$ {\it{r\/}})
\end{tabbing}
\begin{verbatim};;;;;;;;;;;;;;;;;common in mole and atom end.
;;;;;;;;;;;;;;;;;Lemmas on hints.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-hint-in-l8-12 \\ \index{ex-hint-in-l8-12}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-in-g34 \\ \index{ex-hint-in-g34}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-l-g-h \\ \index{ex-hint-l-g-h}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-lp-gp-h-in-int-8-12-3-4 \\ \index{ex-hint-lp-gp-h-in-int-8-12-3-4}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-lp-gp-h-leq-h-j \\ \index{ex-hint-lp-gp-h-leq-h-j}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-not-in-g02 \\ \index{ex-hint-not-in-g02}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{ }}{\tt{2}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  hint-wtn \\ \index{hint-wtn}
\=\+(({\it{r\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{intersect-8-12-3-4-at-n}}\,({\it{r\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n} $\wedge$ ({\it{r\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})) \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;;;;;;;;;;;;;;;Lemmas on hints end.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-k-in-l7-imp \\ \index{lm-k-in-l7-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-neq-k-imp \\ \index{ex-hint-neq-k-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-neq-k-in-l7 \\ \index{ex-hint-neq-k-in-l7}
\=\+({\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} $\wedge$ {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}) \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-in-int-8-12-3-4-l7 \\ \index{ex-hint-in-int-8-12-3-4-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-wtn-l7 \\ \index{ex-hint-wtn-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1b-k-in-l7-imp \\ \index{b1b-k-in-l7-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-j-leq-k \\ \index{h-j-leq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-lp8-then-k-in-g34 \\ \index{lm-lp8-then-k-in-g34}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}


\noindent{\sc Event}:   Disable lm-lp8-then-k-in-g34.

\addvspace{18pt}

\begin{tabbing}{\sc Theorem}:  lp8-then-k-in-g34 \\ \index{lp8-then-k-in-g34}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-k-in-g34 \\ \index{lm-k-in-g34}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-g34 \\ \index{k-in-g34}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-int \\ \index{k-in-int}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{k\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l5-imp \\ \index{k-in-l5-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{verbatim};;;This is slow.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-hint-in-l12 \\ \index{ex-hint-in-l12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{12}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  r-neq-k-l8-11-k-in-lp8-12 \\ \index{r-neq-k-l8-11-k-in-lp8-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{r\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{r\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  r-eq-k-l8-11-k-in-lp8-12 \\ \index{r-eq-k-l8-11-k-in-lp8-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l8-11-k-in-lp8-12 \\ \index{l8-11-k-in-lp8-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{r\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{r\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  hint-in-l8-11 \\ \index{hint-in-l8-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-not-in-l12 \\ \index{ex-hint-not-in-l12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{12}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-in-int-8-12-3-4-l8-11 \\ \index{ex-hint-in-int-8-12-3-4-l8-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-wtn-l8-11 \\ \index{ex-hint-wtn-l8-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l8-11-imp \\ \index{k-in-l8-11-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-lp9-12-k-in-l8-11 \\ \index{m-lp9-12-k-in-l8-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-lp9-12-imp \\ \index{k-in-lp9-12-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-not-in-l7-imp \\ \index{k-not-in-l7-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm1-b1b-i-eq-k-j-neq-k \\ \index{lm1-b1b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-lp-gp-h-leq-hp-j \\ \index{ex-hint-lp-gp-h-leq-hp-j}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-neq-k-then-hp-eq-h \\ \index{j-neq-k-then-hp-eq-h}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b1b-i-eq-k-j-neq-k \\ \index{lm-b1b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{j\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1b-i-eq-k-j-neq-k \\ \index{b1b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1b-i-j-eq-k \\ \index{b1b-i-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{k\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{b1b}\-
\end{tabbing}

\begin{verbatim};;;I wonder if (b1d n l h i) is 
;;;better than (b1d n l h k).
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b1b-i-eq-k \\ \index{b1b-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b1b}\-
\end{tabbing}

\begin{verbatim};;;I wonder why the following two do not imply 
;;;lm-b1b-i-neq-k-j-eq-k.
;;;(prove-lemma k-not-in-l3 (rewrite)
;;;   (implies (and (molws n l g h)
;;;                 (member i (nset n))
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (not (equal i k))
;;;                 (lg n l g)
;;;                 (at lp k 3)
;;;                 (not (at l k 3))
;;;                 (union-at-n l i '(8 9 10 11 12)))
;;;         (exist-hint-8-12-3-4 n lp gp hp k)))
;;;
;;;(prove-lemma k-in-l3 (rewrite)
;;;   (implies (and (molws n l g h)
;;;                 (member i (nset n))
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (at l k 3)
;;;                 (at lp k 3)
;;;                 (union-at-n l i '(8 9 10 11 12)))
;;;         (exist-hint-8-12-3-4 n lp gp hp k)))
;;;
;;;(prove-lemma lm-b1b-i-neq-k-j-eq-k (rewrite)
;;;   (implies (and (molws n l g h)
;;;                 (member i (nset n))
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (not (equal i k))
;;;                 (lg n l g)
;;;                 (at lp k 3)
;;;                 (union-at-n l i '(8 9 10 11 12)))
;;;         (exist-hint-8-12-3-4 n lp gp hp k)))

\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-hint-leq-h-k \\ \index{ex-hint-leq-h-k}
\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-leq-sub1-ex-hint \\ \index{h-k-leq-sub1-ex-hint}
\=\+(\=\+{\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth} $\not=$ {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} $-\;1$) $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}



\begin{tabbing}{\sc Theorem}:  ex-hint-neq-h-k \\ \index{ex-hint-neq-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth} $\not=$ {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-hp-k-leq-ex-l-g-h \\ \index{lm-hp-k-leq-ex-l-g-h}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} $-\;1$) $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-cond-l3 \\ \index{ex-cond-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;$(({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} $-\;1$) $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  hp-k-leq-ex-l-g-h \\ \index{hp-k-leq-ex-l-g-h}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-hint-neq-k-in-l3 \\ \index{ex-hint-neq-k-in-l3}
\=\+(\=\+{\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not=$ {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\-
\end{tabbing}

\begin{verbatim};;;This is successfully proved 
;;;by m-gp-same-g and m-lp-same-l.
;;;This is successfully proved by ex-hint-neq-k-imp, 
;;;ex-hint-neq-k-in-l3 and ex-hint-in-l8-12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  ex-hint-l-g-h-in-int-8-12-3-4 \\ \index{ex-hint-l-g-h-in-int-8-12-3-4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  ex-l-g-h-k-in-l3 \\ \index{ex-l-g-h-k-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-k-in-l3 \\ \index{lm-k-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l3 \\ \index{k-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  hp-k-leq-i \\ \index{hp-k-leq-i}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1b-u-neq-k \\ \index{b1b-u-neq-k}
\=\+(\=\+({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{k\/}} $\not=$ {\it{u\/}})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-u-in-int-8-12-3-4 \\ \index{lm-u-in-int-8-12-3-4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{u\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{u\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-neq-u-in-lp8-12 \\ \index{k-neq-u-in-lp8-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{u\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{u\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm1-u-in-int-8-12-3-4 \\ \index{lm1-u-in-int-8-12-3-4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{u\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  u-in-int-8-12-3-4 \\ \index{u-in-int-8-12-3-4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{u\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{u\/}}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{u\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{verbatim};;;I wonder why the following does not trigger 
;;;molws-ln-lp, molws-ln-gp.
;;;(prove-lemma h-i-in-g34-imp (rewrite)
;;;   (implies (and (molws n l g h)
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (member (nth h i) (nset n))
;;;                 (lg n l g)
;;;                 (at l k 2)
;;;                 (at lp k 3)
;;;                 (at g (nth h i) 4))
;;;            (exist-hint-8-12-3-4 n lp gp hp k)))
;;;although
;;;(prove-lemma h-i-in-g34-imp (rewrite)
;;;   (implies (and (member (nth h i) (nset n))
;;;                 (molws n l g h)
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (lg n l g)
;;;                 (at l k 2)
;;;                 (at lp k 3)
;;;                 (at g (nth h i) 4))
;;;            (exist-hint-8-12-3-4 n lp gp hp k)))
;;;does.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  h-i-in-g34-imp \\ \index{h-i-in-g34-imp}
\=\+(\=\+({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-not-in-g34 \\ \index{i-not-in-g34}
\=\+(\=\+($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-in-int-8-12-3-4 \\ \index{i-in-int-8-12-3-4}
\=\+(\=\+{\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{i\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-in-g34 \\ \index{i-in-g34}
\=\+(\=\+{\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l2 \\ \index{k-in-l2}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lp3-then-l3-or-l2 \\ \index{lp3-then-l3-or-l2}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{2}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-k-not-in-l3 \\ \index{lm-k-not-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{3}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-not-in-l3 \\ \index{k-not-in-l3}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{3}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1b-i-neq-k-j-eq-k \\ \index{b1b-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{k\/}})\index{b1b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-i-neq-k-in-int-8-12-3-4 \\ \index{lm-i-neq-k-in-int-8-12-3-4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{i\/}}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{i\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-neq-k-in-int-8-12-3-4 \\ \index{i-neq-k-in-int-8-12-3-4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{intersect-8-12-3-4-at-n}}\,({\it{i\/}}, {\it{lp\/}}, {\it{gp\/}})\index{intersect-8-12-3-4-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-j-leq-i \\ \index{h-j-leq-i}
\=\+(\=\+{\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b2a})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{i\/}} $\not<$ {\rm{nth}}\,({\it{h\/}}, {\it{j\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-neq-k-ex-hint-in-l12 \\ \index{i-neq-k-ex-hint-in-l12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-neq-k-ex-hint-not-in-l12 \\ \index{i-neq-k-ex-hint-not-in-l12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\tt{12}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm1-b1b-i-j-neq-k \\ \index{lm1-b1b-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b1b-i-j-neq-k \\ \index{lm-b1b-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1b-i-j-neq-k \\ \index{b1b-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1b-i-neq-k \\ \index{b1b-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b}\-
\end{tabbing}


\begin{verbatim};;;I wonder if (b1d n l h i) and 
;;;(b1d n l h j) are better than
;;;(b1d n l h k).
;;;(b1b n l g h (nth h i) j) and (b1b n l g h (nth h k) j).
;;;What about (b2a l (exist-hint-8-12-3-4 n l g h j) i)
;;;           (b3a l g (exist-hint-8-12-3-4 n l g h j) i) ?

\end{verbatim}\begin{tabbing}{\sc Theorem}:  mrho-preserves-b1b \\ \index{mrho-preserves-b1b}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b0b}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b0b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{exist-hint-8-12-3-4}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{j\/}})\index{exist-hint-8-12-3-4}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1b}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b1b}\-
\end{tabbing}


\begin{verbatim};;;;;;;;;;;;;   b1c   ;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;common in mole and atom.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  not-g34-then-not-g4 \\ \index{not-g34-then-not-g4}
($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}) $\rightarrow$ ($\neg$ {\rm{at}}\,({\it{g\/}}, {\it{i\/}}, {\tt{4}})\index{at})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  contra-if4 \\ \index{contra-if4}
\=\+(({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} $\wedge$ {\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}) \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}
\begin{verbatim};;;;;;;;;;;;;;common in mole and atom end.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lp8-not-l5-then-l7 \\ \index{lp8-not-l5-then-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{5}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lp8-not-g34-then-k-in-l7 \\ \index{lp8-not-g34-then-k-in-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-k-in-l7 \\ \index{lm-k-in-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l7 \\ \index{k-in-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-cond-l7 \\ \index{h-k-cond-l7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$({\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth} $=$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-h-k-g4 \\ \index{lm-h-k-g4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  h-k-g4 \\ \index{h-k-g4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm1-i-eq-k-then-h-k-neq-k \\ \index{lm1-i-eq-k-then-h-k-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}, {\tt{4}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;Need k-in-l7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-i-eq-k-then-h-k-neq-k \\ \index{lm-i-eq-k-then-h-k-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{7}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  i-eq-k-then-h-k-neq-k \\ \index{i-eq-k-then-h-k-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1c-i-eq-k-hp-k-neq-k \\ \index{b1c-i-eq-k-hp-k-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{gp\/}}, {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1c-i-eq-k \\ \index{b1c-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}})\index{b1c}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l9-11-then-in-lp9-12 \\ \index{l9-11-then-in-lp9-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-lp9-12 \\ \index{k-in-lp9-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}


\begin{tabbing}{\sc Theorem}:  lm-k-not-in-l12-imp \\ \index{lm-k-not-in-l12-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-not-in-l12-imp \\ \index{k-not-in-l12-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{k\/}}, {\tt{4}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-not-in-l12 \\ \index{k-not-in-l12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{k\/}}, {\it{i\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{12}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm1-b1c-i-neq-k-h-i-eq-k \\ \index{lm1-b1c-i-neq-k-h-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\it{i\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b1c-i-neq-k-h-i-eq-k \\ \index{lm-b1c-i-neq-k-h-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\it{i\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{gp\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3a-h-rholemma \\ \index{b3a-h-rholemma}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{nth}}\,({\it{hp\/}}, {\it{i\/}})\index{nth}, {\it{i\/}})\index{b3a}\-
\end{tabbing}

\begin{verbatim};;;m-l-same-lp and m-gp-same-g are used.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b1c-i-neq-k-h-i-eq-k \\ \index{b1c-i-neq-k-h-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}})\index{b1c}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b1c-i-h-i-neq-k \\ \index{lm-b1c-i-h-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{3}}{\tt{ }}{\tt{4}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\wedge$ {\rm{at}}\,({\it{gp\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\tt{4}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1c-i-h-i-neq-k \\ \index{b1c-i-h-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}})\index{b1c}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1c-i-neq-k \\ \index{b1c-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}})\index{b1c}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrho-preserves-b1c \\ \index{mrho-preserves-b1c}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}})\index{b1c} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}, {\it{i\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1c}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}})\index{b1c}\-
\end{tabbing}


\begin{verbatim};;;;;;;;;;;;;   b1d   ;;;;;;;;;;;;;;;;

\end{verbatim}\begin{tabbing}{\sc Theorem}:  remainder-quotient \\ \index{remainder-quotient}
({\it{x\/}} ${}${\bf{mod}}${}$ ($1\;+$ {\it{x\/}})) $=$ {\rm{fix}}\,({\it{x\/}})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm1-member-remainder \\ \index{lm1-member-remainder}
({\it{x\/}} $\not<$ {\it{n\/}}) $\rightarrow$ (($1\;+$ {\it{x\/}}) $\not\in$ {\rm{nset}}\,({\it{n\/}} $-\;1$)\index{nset})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-member-remainder \\ \index{lm-member-remainder}
(($1\;+$ {\it{x\/}}) $\in$ {\rm{nset}}\,({\it{n\/}} $-\;1$)\index{nset}) $\rightarrow$ (($1\;+$ ({\it{x\/}} ${}${\bf{mod}}${}$ {\it{n\/}})) $\in$ {\rm{nset}}\,({\it{n\/}} $-\;1$)\index{nset})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  member-remainder \\ \index{member-remainder}
({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) $\rightarrow$ (($1\;+$ (({\it{j\/}} $-\;1$) ${}${\bf{mod}}${}$ {\it{n\/}})) $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  one-nset \\ \index{one-nset}
({\it{n\/}} $\not\simeq {\tt{0}}$) $\rightarrow$ ({\tt{1}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset})
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b1d-i-eq-k \\ \index{lm-b1d-i-eq-k}
\=\+(\=\+{\rm{listp}}\,({\it{l\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{listp}}\,({\it{h\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{n\/}} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth} $\in\;${\bf{N}}${}$) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{l\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\rm{length}}\,({\it{h\/}})\index{length} $=$ {\it{n\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}})\index{b1d}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1d-i-eq-k \\ \index{b1d-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{k\/}})\index{b1d})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{k\/}})\index{b1d}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b1d-neq-k \\ \index{b1d-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}})\index{b1d})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}})\index{b1d}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrhoi-preserves-b1d \\ \index{mrhoi-preserves-b1d}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{l\/}}, {\it{h\/}}, {\it{i\/}})\index{b1d})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b1d}}\,({\it{n\/}}, {\it{lp\/}}, {\it{hp\/}}, {\it{i\/}})\index{b1d}\-
\end{tabbing}



\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;   b2.ev   ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;   b2a   ;;;;;;;;;;;;;;;;

;* i-eq-k-j-neq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-lt-h-k \\ \index{j-lt-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-case-k-in-l9 \\ \index{lm-case-k-in-l9}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;In the rewrite rule a set of hypotheses is replaced
;;;by anther set of formulas. Thus if in a proof 
;;;intended beforehand there is a formula belonging to 
;;;more than one set of hypotheses which are expected to 
;;;be replaced, Bmp is very likely to be unsuccessful.
;;;If j is not equal to k and the k's entry in l is 
;;;between 10 and 12, then the j's entry in lp is not 
;;;between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  case-k-in-l9 \\ \index{case-k-in-l9}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;need un10-11-then-un10-12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  case-k-in-l10-11 \\ \index{case-k-in-l10-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l10-11-or-l9 \\ \index{k-in-l10-11-or-l9}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b2a-i-eq-k-j-neq-k \\ \index{lm-b2a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;I proved
;;;(prove-lemma lm-b2a-i-eq-k-j-neq-k (rewrite)
;;;   (implies (and (molws n l g h)
;;;                 (member j (nset n))
;;;                 (member k (nset n))
;;;                 (mrhoi n k l g h lp gp hp)
;;;                 (not (equal j k))
;;;                 (lessp j k)
;;;                 (lg n l g)
;;;                 (b2a l k j)
;;;                 (b2b l h k j)
;;;                 (union-at-n lp k '(10 11 12)))
;;;             (not (union-at-n l j '(5 6 7 8 9 10 11 12)))))
;;;and tried to prove the following lemma counting on 
;;;m-lp-same-l, but it was unsuccessful.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  b2a-i-eq-k-j-neq-k \\ \index{b2a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2a}}\,({\it{lp\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a}\-
\end{tabbing}

\begin{verbatim};* i-neq-k-j-eq-k

;;;If the k's entry in l is not 4 and 
;;;the k's entry in lp is between 5 and 7, then
;;;the k's entry in l is between 5 and 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-k-in-lp5-7-not-l4-then-l5-7 \\ \index{m-k-in-lp5-7-not-l4-then-l5-7}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 5 and 7 then 
;;;the k's entry in l is  certainly between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-k-in-lp5-7-then-l5-11 \\ \index{m-k-in-lp5-7-then-l5-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is 8 then k's entry 
;;;in l is either 5 or 7.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-lp8-k-in-l57 \\ \index{m-lp8-k-in-l57}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{7}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is 8,
;;;then the k's entry in l is between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-k-in-lp8-then-l5-11 \\ \index{m-k-in-lp8-then-l5-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{8}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in lp is between 9 and 12,
;;;then the k's entry in l is between 5 and 12.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-k-in-lp9-12-then-l5-11 \\ \index{m-k-in-lp9-12-then-l5-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{verbatim};;;If the k's entry in l is 4 an the k's entry in lp is
;;;between 5 and 12, then the k's entry in l is 
;;;between 5 and 11.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  m-k-in-l5-11 \\ \index{m-k-in-l5-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-k-not-in-l4 \\ \index{m-k-not-in-l4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{4}})\index{at}) \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-k-not-in-lp5-12 \\ \index{m-k-not-in-lp5-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b2a-i-neq-k-j-eq-k \\ \index{lm-b2a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b2a-i-neq-k-j-eq-k \\ \index{b2a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b2a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{k\/}})\index{b2a}\-
\end{tabbing}

\begin{verbatim};* i-j-neq-k-neq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  b2a-i-j-neq-k \\ \index{b2a-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b2a-i-neq-k \\ \index{b2a-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b2a-i-eq-k \\ \index{b2a-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2a}}\,({\it{lp\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrho-preserves-b2a \\ \index{mrho-preserves-b2a}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2a}}\,({\it{lp\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a}\-
\end{tabbing}



\begin{verbatim};;;;;;;;;;;;;   b2b   ;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;Common in atom and mole.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l9-then-un8-12 \\ \index{l9-then-un8-12}
{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}
\begin{verbatim};;;;;;;;;;;;;;Common in atom and mole end.

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lg-nth-h-k \\ \index{lg-nth-h-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l9-g01 \\ \index{l9-g01}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{9}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{0}}{\tt{ }}{\tt{1}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l9-nth-h-k-eq-j \\ \index{l9-nth-h-k-eq-j}
\=\+(\=\+{\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{9}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-j-not-in-l5-12 \\ \index{lm-j-not-in-l5-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cond-l9 \\ \index{cond-l9}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-not-in-l5-12 \\ \index{j-not-in-l5-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  k-in-l9 \\ \index{k-in-l9}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{9}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;The order of the hints if crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b2b-i-eq-k-j-neq-k \\ \index{lm-b2b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b2b-i-eq-k-j-neq-k \\ \index{b2b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2b}}\,({\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b2b-i-eq-k \\ \index{b2b-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2b}}\,({\it{lp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b2b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  not-k-in-l5-12-imp \\ \index{not-k-in-l5-12-imp}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{verbatim};;;The order of hyptheses is crucial.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b2b-i-neq-k-j-eq-k \\ \index{lm-b2b-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b2b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{9}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b2b-i-neq-k-j-eq-k \\ \index{b2b-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2b}}\,({\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{k\/}})\index{b2b}\-
\end{tabbing}

\begin{verbatim};;;The position of (member k (nset n)) is 
;;;crucial to trigger rholemmas.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b2b-i-j-neq-k \\ \index{b2b-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2b}}\,({\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b2b-i-neq-k \\ \index{b2b-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2b}}\,({\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrho-preserves-b2b \\ \index{mrho-preserves-b2b}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\it{i\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b2b}}\,({\it{l\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b2b}}\,({\it{lp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b2b}\-
\end{tabbing}





\begin{verbatim};;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;   b3.ev   ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;   b3a   ;;;;;;;;;;;;;;;;

;* i-neq-k-j-eq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b3a-k-in-l9-11 \\ \index{lm-b3a-k-in-l9-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{k\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3a-k-in-l9-11 \\ \index{b3a-k-in-l9-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{k\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-k-in-lp9-12 \\ \index{m-k-in-lp9-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b3a-i-neq-k-j-eq-k \\ \index{lm-b3a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{k\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3a-i-neq-k-j-eq-k \\ \index{b3a-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{k\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{i\/}}, {\it{k\/}})\index{b3a}\-
\end{tabbing}

\begin{verbatim};* i-eq-k-j-neq-k
\end{verbatim}\begin{tabbing}{\sc Theorem}:  cond-lp12 \\ \index{cond-lp12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3a-j-in-l5-12 \\ \index{b3a-j-in-l5-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  m-k-in-l11 \\ \index{m-k-in-l11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b3a-i-eq-k-j-neq-k \\ \index{lm-b3a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{12}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3a-i-eq-k-j-neq-k \\ \index{b3a-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{k\/}}, {\it{j\/}})\index{b3a}\-
\end{tabbing}

\begin{verbatim};* i-j-neq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  b3a-i-j-neq-k \\ \index{b3a-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{j\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{i\/}}, {\it{j\/}})\index{b3a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3a-i-neq-k \\ \index{b3a-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{j\/}})\index{b3a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{i\/}}, {\it{j\/}})\index{b3a}\-
\end{tabbing}

\begin{verbatim};* i-j-eq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  b3a-i-j-eq-k \\ \index{b3a-i-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{k\/}}, {\it{k\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{k\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{k\/}}, {\it{k\/}})\index{b3a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3a-i-eq-k \\ \index{b3a-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{k\/}}, {\it{j\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{k\/}}, {\it{j\/}})\index{b3a}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrho-preserves-b3a \\ \index{mrho-preserves-b3a}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3a}}\,({\it{l\/}}, {\it{g\/}}, {\it{i\/}}, {\it{j\/}})\index{b3a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3a}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{i\/}}, {\it{j\/}})\index{b3a}\-
\end{tabbing}


\begin{verbatim};;;;;;;;;;;;;   b3b   ;;;;;;;;;;;;;;;;

;;;;;;;;;;;;;;common in atom and mole.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  l10-then-un10-12 \\ \index{l10-then-un10-12}
{\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l11-then-un9-12 \\ \index{l11-then-un9-12}
{\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l11-then-un8-12 \\ \index{l11-then-un8-12}
{\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} $\rightarrow$ {\rm{union-at-n}}\,({\it{l\/}}, {\it{i\/}}, {\tt{'}}{\tt{(}}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n}
\end{tabbing}
\begin{verbatim};;;;;;;;;;;;;;common in atom and mole end.

;* i-neq-k-j-eq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b3b-k-in-l9-11 \\ \index{lm-b3b-k-in-l9-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3b-k-in-l9-11 \\ \index{b3b-k-in-l9-11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{)}})\index{union-at-n}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-b3b-i-neq-k-j-eq-k \\ \index{lm-b3b-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{lp\/}}, {\it{gp\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{i\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $<$ {\rm{nth}}\,({\it{h\/}}, {\it{i\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{gp\/}}, {\it{k\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3b-i-neq-k-j-eq-k \\ \index{b3b-i-neq-k-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{k\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{k\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3b}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{k\/}})\index{b3b}\-
\end{tabbing}

\begin{verbatim};* i-eq-k-j-neq-k

\end{verbatim}\begin{tabbing}{\sc Theorem}:  j-in-g4 \\ \index{j-in-g4}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l11-g14 \\ \index{l11-g14}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at})\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{union-at-n}}\,({\it{g\/}}, {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}, {\tt{'}}{\tt{(}}{\tt{2}}{\tt{ }}{\tt{3}}{\tt{)}})\index{union-at-n})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  l11-nth-h-k-eq-j \\ \index{l11-nth-h-k-eq-j}
\=\+(\=\+{\rm{at}}\,({\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lm-j-in-l5-12 \\ \index{lm-j-in-l5-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  cond-l11 \\ \index{cond-l11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$(({\it{j\/}} $-\;1$) $<$ {\rm{nth}}\,({\it{h\/}}, {\it{k\/}})\index{nth})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-in-l5-12 \\ \index{j-in-l5-12}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  j-leq-add1k-then-k-not-in-l10 \\ \index{j-leq-add1k-then-k-not-in-l10}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ ($1\;+$ {\it{k\/}})))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  not-j-leq-add1k-then-k-not-in-l10 \\ \index{not-j-leq-add1k-then-k-not-in-l!10}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not<$ ($1\;+$ {\it{k\/}})))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at})\-
\end{tabbing}

\begin{verbatim};;;The order of (member k (nset n)) and 
;;;(member j (nset n)) are switched deliberately.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  k-not-in-l10 \\ \index{k-not-in-l10}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at})\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  lp11-then-l11-or-l10 \\ \index{lp11-then-l11-or-l10}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$($\neg$ {\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{10}})\index{at}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;When the order of (member j (nset n)) and 
;;;(member k (nset n)) is switched, the order of
;;;hints must be switched, in order to make the proof 
;;;successful.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  b3b-k-in-l11 \\ \index{b3b-k-in-l11}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}))\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{l\/}}, {\it{k\/}}, {\tt{11}})\index{at}\-
\end{tabbing}

\begin{verbatim};;;When the order of (member j (nset n)) and 
;;;(member k (nset n)) is switched then the order of
;;;hints must be switched in order to make the proof 
;;;successful.
\end{verbatim}\begin{tabbing}{\sc Theorem}:  lm-b3b-i-eq-k-j-neq-k \\ \index{lm-b3b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{at}}\,({\it{lp\/}}, {\it{k\/}}, {\tt{11}})\index{at} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $<$ {\rm{nth}}\,({\it{hp\/}}, {\it{k\/}})\index{nth}) \\ 
$\wedge$$\;\;\;\;${\rm{union-at-n}}\,({\it{l\/}}, {\it{j\/}}, {\tt{'}}{\tt{(}}{\tt{5}}{\tt{ }}{\tt{6}}{\tt{ }}{\tt{7}}{\tt{ }}{\tt{8}}{\tt{ }}{\tt{9}}{\tt{ }}{\tt{10}}{\tt{ }}{\tt{11}}{\tt{ }}{\tt{12}}{\tt{)}})\index{union-at-n})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{at}}\,({\it{g\/}}, {\it{j\/}}, {\tt{4}})\index{at}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3b-i-eq-k-j-neq-k \\ \index{b3b-i-eq-k-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3b}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3b-i-j-neq-k \\ \index{b3b-i-j-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3b}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3b-i-neq-k \\ \index{b3b-i-neq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\not=$ {\it{k\/}}) \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3b}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3b-i-j-eq-k \\ \index{b3b-i-j-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{k\/}})\index{b3b})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3b}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{k\/}})\index{b3b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  b3b-i-eq-k \\ \index{b3b-i-eq-k}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{k\/}}, {\it{j\/}})\index{b2a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3b}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{k\/}}, {\it{j\/}})\index{b3b}\-
\end{tabbing}

\begin{tabbing}{\sc Theorem}:  mrho-preserves-b3b \\ \index{mrho-preserves-b3b}
\=\+(\=\+{\rm{molws}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}})\index{molws} \\ 
$\wedge$$\;\;\;\;$({\it{i\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{j\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;$({\it{k\/}} $\in$ {\rm{nset}}\,({\it{n\/}})\index{nset}) \\ 
$\wedge$$\;\;\;\;${\rm{mrhoi}}\,({\it{n\/}}, {\it{k\/}}, {\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}})\index{mrhoi} \\ 
$\wedge$$\;\;\;\;${\rm{lg}}\,({\it{n\/}}, {\it{l\/}}, {\it{g\/}})\index{lg} \\ 
$\wedge$$\;\;\;\;${\rm{b1a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b1a} \\ 
$\wedge$$\;\;\;\;${\rm{b3b}}\,({\it{l\/}}, {\it{g\/}}, {\it{h\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b} \\ 
$\wedge$$\;\;\;\;${\rm{b2a}}\,({\it{l\/}}, {\it{i\/}}, {\it{j\/}})\index{b2a})\- \\ 
$\rightarrow$$\;\;\;\;${\rm{b3b}}\,({\it{lp\/}}, {\it{gp\/}}, {\it{hp\/}}, {\it{i\/}}, {\it{j\/}})\index{b3b}\-
\end{tabbing}


%\end{list}
\printindex
\end{document}
