<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">PERM</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span><br/>  for <font color="Olive" title="b2">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>  st  not <font color="Olive" title="b2">perm</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; (  for <font color="Olive" title="b3">p</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Olive" title="b1">PERM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b2">perm</font> ) holds <br/><font color="Olive" title="b1">PERM</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">p</font> where <font color="Olive" title="b3">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b3">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">q</font> where <font color="Olive" title="b3">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b3">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">P</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c3">PERM</font> be   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">perm</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>  st  not <font color="Olive" title="b1">perm</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; (  for <font color="Olive" title="b2">p</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c3">PERM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">perm</font> ) holds <br/><font color="Maroon" title="c3">PERM</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">q</font> where <font color="Olive" title="b2">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b2">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">perm</font> be    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">perm</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  &amp; (  for <font color="Olive" title="b1">p</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c3">PERM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">perm</font> ) implies <font color="Maroon" title="c3">PERM</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
 not <font color="Maroon" title="c4">perm</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">and </span><br/><a NAME="E2:64"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
 for <font color="Olive" title="b1">p</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> holds  <font color="Maroon" title="c3">PERM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">perm</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">PERM</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">E</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">OD</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> ;<br/>
<a NAME="E3:64"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">q</font> where <font color="Olive" title="b2">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b2">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  iff  ex <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b3">p</font> where <font color="Olive" title="b3">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b3">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  iff  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:64_1"/>
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  implies  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) implies <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c8">perm9</font> = <font color="Maroon" title="c4">perm</font> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  as    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_7.html#T18" title="MATRIX_7:th.18">MATRIX_7:18</a></span>;<br/>
<a NAME="E2:64_1_1"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T52" title="FUNCT_2:th.52">FUNCT_2:52</a></span>;<br/>
<a NAME="E3:64_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="matrix11.html#E29"><span class="lab"><font color="Green" title="E28">Lm3</font></span></a></span>;<br/>
<a NAME="E4:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:64_1_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">q</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E7:64_1_1"/><span class="lab"><font color="Green" title="E60">A5</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 <span class="kw">and </span><br/><a NAME="E8:64_1_1"/><span class="lab"><font color="Green" title="E61">A6</font></span>: 
 not <font color="Maroon" title="c9">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 ;<br/>
<a NAME="E9:64_1_1"/><span class="lab"><font color="Green" title="E62">A7</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T21" title="MATRIX_2:th.21">MATRIX_2:21</a></span>;<br/>
<a NAME="E10:64_1_1"/>
<font color="Maroon" title="c1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E11:64_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c8">perm9</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E4:64_1_1"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="matrix_7.html#T28" title="MATRIX_7:th.28">MATRIX_7:28</a></span>;<br/>
<a NAME="E12:64_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A8</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">perm9</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:64_1_1"><span class="lab"><font color="Green" title="E61">A6</font></span></a>, <a class="ref" href="matrix11.html#T25" target="_self" title="MATRIX11:th.25">Th25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">qp9</font> = <font color="Maroon" title="c9">q</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">perm9</font> as    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T39" title="MATRIX_9:th.39">MATRIX_9:39</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">qp9</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">qp9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Maroon" title="c10">qp9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">qp9</font> )</span><br/>

<a NAME="E14:64_1_1"/><span class="lab"><font color="Green" title="E64">A9</font></span>: 
<font color="Maroon" title="c8">perm9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">perm</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_2.html#T22" title="MATRIX_2:th.22">MATRIX_2:22</a></span>;<br/>
<a NAME="E15:64_1_1"/>
<font color="Maroon" title="c3">PERM</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">qp9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">qp9</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">perm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E57">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:64_1_1"/>
( <font color="Maroon" title="c10">qp9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Maroon" title="c10">qp9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">qp9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:64_1_1"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="#E2:64_1_1"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E14:64_1_1"><span class="lab"><font color="Green" title="E64">A9</font></span></a>, <a class="txt" href="#E9:64_1_1"><span class="lab"><font color="Green" title="E62">A7</font></span></a>, <a class="txt" href="#E12:64_1_1"><span class="lab"><font color="Green" title="E63">A8</font></span></a>, <a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:64_1"/>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">p</font> where <font color="Olive" title="b2">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b2">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:64_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">PERM</font>
 <span class="kw">and </span><br/><a NAME="E5:64_1"/><span class="lab"><font color="Green" title="E58">A10</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E6:64_1"/><span class="lab"><font color="Green" title="E59">A11</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font>
 ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">p</font> being    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font><span class="kw"> such that </span><br/><a NAME="E8:64_1"/><span class="lab"><font color="Green" title="E60">A12</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font>
 <span class="kw">and </span><br/><a NAME="E9:64_1"/><span class="lab"><font color="Green" title="E61">A13</font></span>: 
<font color="Maroon" title="c9">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E58">A10</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">pp</font> = <font color="Maroon" title="c9">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">perm</font> as    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_9.html#T39" title="MATRIX_9:th.39">MATRIX_9:39</a></span>;<br/>
<a NAME="E11:64_1"/><span class="lab"><font color="Green" title="E62">A14</font></span>: 
<font color="Maroon" title="c3">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c4">perm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="#E8:64_1"><span class="lab"><font color="Green" title="E60">A12</font></span></a></span>;<br/>
<a NAME="E12:64_1"/>
 not <font color="Maroon" title="c10">pp</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="#E9:64_1"><span class="lab"><font color="Green" title="E61">A13</font></span></a>, <a class="ref" href="matrix11.html#T25" target="_self" title="MATRIX11:th.25">Th25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:64_1"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E59">A11</font></span></a>, <a class="txt" href="#E11:64_1"><span class="lab"><font color="Green" title="E62">A14</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:64"/>
<font color="Maroon" title="c3">PERM</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> : <font color="Olive" title="b1">p</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is    <a href="matrix_2.html#M1" title="MATRIX_2:mode.1">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> :  not <font color="Olive" title="b1">q</font> is  <a href="matrix_2.html#V5" title="MATRIX_2:attr.5">even</a>  </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
