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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">IT</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>



<b>assume </b><a NAME="E1:33"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">IT</font> is <a href="matrix_2.html#V3">even</a> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<a NAME="E2:33"/><i><font color="Green">E40</font></i>: 
 <a href="matrix_2.html#K11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/>
<b>consider </b><font color="Maroon">l</font> being   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E4:33"/><i><font color="Green">E42</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font> holds <br/> ex <font color="Olive">q</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> st <br/>( <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  ) ) )
 <b>by </b><i><a class="ref" href="matrix_2.html#D15">MATRIX_2:def 15</a>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">lv</font> = <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">l</font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a>  as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<a NAME="E6:33"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E7:33"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lv</font>
 
<b>by </b><i/>;<br/>
<a NAME="E8:33"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lv</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font>;<br/>
<b>reconsider </b><font color="Maroon">ITP</font> = <font color="Maroon">IT</font> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">ITG</font> = <font color="Maroon">ITP</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E11:33"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">ITG</font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">ITP</font> <a href="matrix_7.html#K3" target="_self">"</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:33"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">IT</font> <a href="matrix_7.html#K3" target="_self">"</a>  <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">lv</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:33"/>
for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lv</font> holds <br/> ex <font color="Olive">q2</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> st <br/>( <font color="Maroon">lv</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">q2</font> &amp; <font color="Olive">q2</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lv</font>
 ;<br/>

<a NAME="E2:33_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">lv</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:33_1"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="txt" href="matrix_7.html#E1:33"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:33_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E5:33_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:33_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="txt" href="matrix_7.html#E2:33"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E7:33_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:33_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font>
 
;<br/>
<a NAME="E9:33_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="binarith.html#K3">-'</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  ;<br/>
<b>reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E12:33_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E13:33_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:33"><i><font color="Green">E42</font></i></a>, <a class="txt" href="matrix_7.html#E6:33"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E14:33_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">l</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E16:33_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E8:33"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E17:33_1"/>
<font color="Maroon">lv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lv</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">qq</font> = <font color="Maroon">lv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon">i1</font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j1</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E20:33_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">q</font> holds <br/><font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> ) )
 <b>by </b><i><a class="txt" href="matrix_7.html#E11:33"><i><font color="Green">E46</font></i></a>, <a class="ref" href="matrix_2.html#D14">MATRIX_2:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">qf</font> = <font color="Maroon">q</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E22:33_1"/><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E23:33_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">qf</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">qqf</font> = <font color="Maroon">qq</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>, <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E25:33_1"/><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">qqf</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="matrix_7.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E26:33_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E12:33"><i><font color="Green">E47</font></i></a>, , </i>;<br/>
<a NAME="E27:33_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="matrix_7.html#E8:33"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E28:33_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E11:33"><i><font color="Green">E46</font></i></a>, <a class="txt" href="matrix_7.html#E1:33_1"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E29:33_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">l</font></span>)</span>
 
<b>by </b><i>, , , <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green">E70</font></i>: <font color="Maroon">qq</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">l</font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">l</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ii</font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i>, <a class="txt" href="matrix_7.html#E3:33_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a>, <a class="txt" href="matrix_7.html#E7:33"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_7.html#E8:33"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="matrix_7.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="matrix_7.html#E2:33_1"><i><font color="Green">E53</font></i></a>, , </i>
;<br/>
<a NAME="E31:33_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> holds <br/><font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> )
 ;<br/>

<a NAME="E2:33_1_2"/><b>then </b>
<font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i>, , <a class="txt" href="matrix_7.html#E12:33"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_2"/>
<font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E5:33_1"><i><font color="Green">E57</font></i></a>, , , <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:33_1"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, <a class="txt" href="matrix_7.html#E8:33"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E33:33_1"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E11:33"><i><font color="Green">E46</font></i></a>, <a class="txt" href="matrix_7.html#E5:33_1"><i><font color="Green">E57</font></i></a></i>;<br/>
<i><font color="Green">E101</font></i>: <font color="Maroon">qq</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">l</font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">l</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">ii</font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i>, <a class="txt" href="matrix_7.html#E3:33_1"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a>, <a class="txt" href="matrix_7.html#E7:33"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_7.html#E8:33"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<font color="Maroon">q</font> <a href="matrix_7.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="matrix_7.html#E8:33_1"><i><font color="Green">E58</font></i></a>, , </i>
;<br/>
<a NAME="E35:33_1"/><i><font color="Green">E102</font></i>: 
<font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E12:33"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a>, <a class="txt" href="matrix_7.html#E12:33_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E36:33_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> &amp; <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> &amp; <font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font> &amp; <font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Olive">i</font> &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i</font> &amp; <font color="Olive">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">j</font> &amp; <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> holds <br/><font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">k</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon">i</font> = <font color="Maroon">i1</font>;<br/>

<b>take </b><font color="Maroon">j</font> = <font color="Maroon">j1</font>;<br/>

<a NAME="E1:33_1_4"/>
<font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E12:33"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a>, <a class="txt" href="matrix_7.html#E12:33_1"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:33_1_4"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i>, <a class="txt" href="matrix_7.html#E12:33"><i><font color="Green">E47</font></i></a>, <a class="txt" href="matrix_7.html#E14:33_1"><i><font color="Green">E60</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E3:33_1_4"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_4"/>
( <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">qq</font> implies <font color="Maroon">qq</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:33_1"><i><font color="Green">E56</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E37:33_1"/><b>then </b>
<font color="Maroon">qq</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a> 
 
<b>by </b><i><a class="ref" href="matrix_2.html#D14">MATRIX_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E38:33_1"/>
 ex <font color="Olive">q2</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font> st <br/>( <font color="Maroon">lv</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">q2</font> &amp; <font color="Olive">q2</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:33"/>
<font color="Maroon">IT</font> <a href="matrix_7.html#K3" target="_self">"</a>  is <a href="matrix_2.html#V3">even</a>
 <b>by </b><i><a class="ref" href="matrix_2.html#D15">MATRIX_2:def 15</a>, , , <a class="txt" href="matrix_7.html#E1"><i><font color="Green">Lemma34</font></i></a>, <a class="ref" href="matrix_7.html#T1" target="_self">Th1</a></i>;<br/>


</div>
