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

<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E57">A1</font></i>: 
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c4">P</font> =  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/>
<b>consider </b><font color="Maroon" title="c5">tr</font> being    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E3:65"/><i><font color="Green" title="E58">A2</font></i>: 
( <font color="Maroon" title="c5">tr</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  &amp; <font color="Maroon" title="c5">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:65"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="matrix11.html#T16" target="_self" title="MATRIX11:th.16">Th16</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">PE</font> being   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font><b> such that </b><br/><a NAME="E5:65"/><i><font color="Green" title="E59">A3</font></i>: 
for <font color="Olive" title="b1">p</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">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="c6">PE</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="c5">tr</font>
 <b>by </b><i><a class="ref" href="matrix11.html#T44" target="_self" title="MATRIX11:th.44">Th44</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">EVEN</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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c8">ODD</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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span> ;<br/>
<a NAME="E6:65"/><i><font color="Green" title="E60">A4</font></i>: 
<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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:65_1"/>
<font color="Maroon" title="c9">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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span> 
 ;<br/>

<a NAME="E2:65_1"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Olive" title="b1">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 
;<br/>
<b>hence </b><a NAME="E3:65_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:65"/><i><font color="Green" title="E61">A5</font></i>: 
<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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:65_2"/><i><font color="Green" title="E62">A6</font></i>: 
<font color="Maroon" title="c9">x</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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span> 
 ;<br/>

<a NAME="E2:65_2"/>
 ex <font color="Olive" title="b1">q</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; not <font color="Olive" title="b1">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E1:65_2"><i><font color="Green" title="E62">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:65_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:65"/>
 <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T30" title="MATRIX_2:th.30">MATRIX_2:30</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">O</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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span> , <font color="Maroon" title="c10">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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span>  as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="matrix11.html#E6:65"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix11.html#E7:65"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="finset_1.html#T13" title="FINSET_1:th.13">FINSET_1:13</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">O</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c10">E</font>
;<br/>

<b>thus </b><a NAME="E10:65"/>
( <font color="Maroon" title="c10">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span>  &amp; <font color="Maroon" title="c9">O</font> <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#M3" title="MATRIX_2:mode.3">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#V3" title="MATRIX_2:attr.3">even</a> </span>}</span>  )
 ;<br/>

<b>thus </b><a NAME="E11:65"/>
<font color="Maroon" title="c10">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">O</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_3"/>
<font color="Maroon" title="c10">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">O</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:65_3"/><i><font color="Green" title="E62">A7</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c9">O</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1" title="XBOOLE_0:def.1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E4:65_3"/>
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">O</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E3:65_3"><i><font color="Green" title="E62">A7</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:65_3"/><b>then </b>
(  ex <font color="Olive" title="b1">p</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Olive" title="b1">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) &amp;  ex <font color="Olive" title="b1">q</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; not <font color="Olive" title="b1">q</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> ) )
 
;<br/>
<b>hence </b><a NAME="E6:65_3"/>
contradiction
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E12:65"/>
<font color="Maroon" title="c10">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">O</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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_4"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:65_4"/>
<font color="Maroon" title="c10">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">O</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="matrix11.html#E6:65"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix11.html#E7:65"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:65_4"/><i><font color="Green" title="E62">A8</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c12">p</font> = <font color="Maroon" title="c11">x</font> as    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="txt" href="matrix11.html#E2:65_4"><i><font color="Green" title="E62">A8</font></i></a></i>;<br/>
<a NAME="E4:65_4"/>
( <font color="Maroon" title="c12">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> or  not <font color="Maroon" title="c12">p</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> )
 
;<br/>
<a NAME="E5:65_4"/><b>then </b>
( <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font> or <font color="Maroon" title="c12">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">O</font> )
 
;<br/>
<b>hence </b><a NAME="E6:65_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">O</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<a NAME="E13:65"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix11.html#NK1" title="MATRIX11:NK.1">2Set</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix11.html#E1:65"><i><font color="Green" title="E57">A1</font></i></a>, <a class="ref" href="matrix11.html#T1" target="_self" title="MATRIX11:th.1">Th1</a></i>;<br/>
<a NAME="E14:65"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<b>by </b><i><a class="ref" href="matrix11.html#T2" target="_self" title="MATRIX11:th.2">Th2</a>, <a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">n2</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 2 as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="ref" href="nat_1.html#T21" title="NAT_1:th.21">NAT_1:21</a></i>;<br/>
<a NAME="E16:65"/>
<font color="Maroon" title="c5">tr</font> is    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
;<br/>
<a NAME="E17:65"/><b>then </b>
not <font color="Maroon" title="c5">tr</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a>
 
<b>by </b><i><a class="txt" href="matrix11.html#E3:65"><i><font color="Green" title="E58">A2</font></i></a>, <a class="ref" href="matrix11.html#T27" target="_self" title="MATRIX11:th.27">Th27</a></i>;<br/>
<a NAME="E18:65"/><b>then </b><i><font color="Green" title="E62">A9</font></i>: 
<font color="Maroon" title="c6">PE</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <font color="Maroon" title="c10">E</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">O</font>
 
<b>by </b><i><a class="txt" href="matrix11.html#E5:65"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="matrix11.html#T47" target="_self" title="MATRIX11:th.47">Th47</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">PERM</font> = <font color="Maroon" title="c6">PE</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font>;<br/>
<a NAME="E19:65"/><i><font color="Green" title="E63">A10</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">PE</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>
 
<b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a></i>;<br/>
<a NAME="E20:65"/><b>then </b><i><font color="Green" title="E64">A11</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">PE</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">PE</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">O</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E6:65"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix11.html#E18:65"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">PERM</font> = <font color="Maroon" title="c6">PE</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">E</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c10">E</font>,<font color="Maroon" title="c9">O</font> <b>by </b><i><a class="ref" href="funct_2.html#T3" title="FUNCT_2:th.3">FUNCT_2:3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c13">PERM</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c5">tr</font>
;<br/>

<a NAME="E22:65"/>
( <font color="Maroon" title="c13">PERM</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c13">PERM</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E20:65"><i><font color="Green" title="E64">A11</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/>
<b>hence </b><a NAME="E23:65"/>
( <font color="Maroon" title="c5">tr</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  &amp; <font color="Maroon" title="c5">tr</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">j</font> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c13">PERM</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">E</font> &amp; <font color="Maroon" title="c13">PERM</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:65"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="matrix11.html#E6:65"><i><font color="Green" title="E60">A4</font></i></a>, <a class="txt" href="matrix11.html#E19:65"><i><font color="Green" title="E63">A10</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>

<b>let </b><font color="Maroon" title="c14">p</font> be    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <font color="Maroon" title="c1">n</font>;<br/>

<b>assume </b><a NAME="E24:65"/><i><font color="Green" title="E65">A12</font></i>: 
<font color="Maroon" title="c14">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E</font>
 ;<br/>

<a NAME="E25:65"/>
( <font color="Maroon" title="c13">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">PE</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p</font> &amp; <font color="Maroon" title="c6">PE</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">tr</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E5:65"><i><font color="Green" title="E59">A3</font></i></a>, <a class="txt" href="matrix11.html#E20:65"><i><font color="Green" title="E64">A11</font></i></a>, <a class="txt" href="matrix11.html#E24:65"><i><font color="Green" title="E65">A12</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E26:65"/>
<font color="Maroon" title="c13">PERM</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">tr</font>
 ;<br/>


</div>
