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

<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">IT</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="E1:33"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c2">IT</font> is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a> &amp; <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 ;<br/>

<a NAME="E2:33"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="matrix_2.html#K11" title="MATRIX_2:func.11">len</a> <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 href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T20" title="MATRIX_2:th.20">MATRIX_2:20</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">l</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span><b> such that </b><br/><a NAME="E4:33"/><i><font color="Green" title="E30">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c3">l</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">l</font> holds <br/> ex <font color="Olive" title="b2">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="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b2">q</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  ) ) )
 <b>by </b><i><a class="ref" href="matrix_2.html#D15" title="MATRIX_2:def.15">MATRIX_2:def 15</a>, <a class="txt" href="matrix_7.html#E1:33"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">lv</font> = <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a>  as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span> ;<br/>
<a NAME="E6:33"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">l</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E7:33"/><b>then </b><i><font color="Green" title="E32">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">lv</font>
 
<b>by </b><i><a class="ref" href="matrix_7.html#D4" target="_self" title="MATRIX_7:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">m</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">ITP</font> = <font color="Maroon" title="c2">IT</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> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">ITG</font> = <font color="Maroon" title="c6">ITP</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E10:33"/>
<font color="Maroon" title="c7">ITG</font> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">ITP</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> 
 
<b>by </b><i><a class="ref" href="matrix_7.html#T27" target="_self" title="MATRIX_7:th.27">Th27</a>, <a class="txt" href="matrix_7.html#E1:33"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E11:33"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c4">lv</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:33"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="matrix_7.html#T26" target="_self" title="MATRIX_7:th.26">Th26</a></i>;<br/>
<a NAME="E12:33"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">lv</font> holds <br/> ex <font color="Olive" title="b2">q2</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="Maroon" title="c4">lv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q2</font> &amp; <font color="Olive" title="b2">q2</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">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" title="c8">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E34">A7</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">lv</font>
 ;<br/>

<a NAME="E2:33_1"/><b>then </b>
<font color="Maroon" title="c8">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">lv</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:33_1"/><b>then </b><i><font color="Green" title="E35">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">i</font> &amp; <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:33"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:33_1"/><b>then </b><i><font color="Green" title="E36">A9</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="xreal_1.html#T50" title="XREAL_1:th.50">XREAL_1:50</a></i>;<br/>
<a NAME="E5:33_1"/><b>then </b><i><font color="Green" title="E37">A10</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:33_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a>, <a class="txt" href="matrix_7.html#E3:33_1"><i><font color="Green" title="E35">A8</font></i></a></i>;<br/>
<a NAME="E7:33_1"/><b>then </b><i><font color="Green" title="E38">A11</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:33_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:33_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">j</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c8">i</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <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>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">ii</font> = <font color="Maroon" title="c8">i</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E11:33_1"/><i><font color="Green" title="E39">A12</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<a NAME="E12:33_1"/>
<font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:33_1"><i><font color="Green" title="E37">A10</font></i></a>, <a class="txt" href="matrix_7.html#E7:33_1"><i><font color="Green" title="E38">A11</font></i></a></i>;<br/>
<a NAME="E13:33_1"/><b>then </b><i><font color="Green" title="E40">A13</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">l</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">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><b> such that </b><br/><a NAME="E15:33_1"/><i><font color="Green" title="E41">A14</font></i>: 
( <font color="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Maroon" title="c11">q</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:33"><i><font color="Green" title="E30">A3</font></i></a></i>;<br/>
<a NAME="E16:33_1"/>
<font color="Maroon" title="c4">lv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">lv</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a>, <a class="txt" href="matrix_7.html#E1:33_1"><i><font color="Green" title="E34">A7</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">qq</font> = <font color="Maroon" title="c4">lv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</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="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">i1</font>, <font color="Maroon" title="c14">j1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:33_1"/><i><font color="Green" title="E42">A15</font></i>: 
( <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Maroon" title="c13">i1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j1</font> &amp; <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">j1</font> &amp; <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">i1</font> &amp; ( for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">i1</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j1</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c11">q</font> holds <br/><font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">k</font> ) )
 <b>by </b><i><a class="txt" href="matrix_7.html#E15:33_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">qf</font> = <font color="Maroon" title="c11">q</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E21:33_1"/><i><font color="Green" title="E43">A16</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</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> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrix_7.html#E1:33"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E22:33_1"/><b>then </b><i><font color="Green" title="E44">A17</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c15">qf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">qqf</font> = <font color="Maroon" title="c12">qq</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> <b>by </b><i><a class="ref" href="matrix_2.html#D11" title="MATRIX_2:def.11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E24:33_1"/><i><font color="Green" title="E45">A18</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c16">qqf</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="txt" href="matrix_7.html#E21:33_1"><i><font color="Green" title="E43">A16</font></i></a></i>;<br/>
<a NAME="E25:33_1"/><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a>, <a class="txt" href="matrix_7.html#E13:33_1"><i><font color="Green" title="E40">A13</font></i></a></i>;<br/>
<a NAME="E26:33_1"/><i><font color="Green" title="E47">A20</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">l</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E6:33"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="matrix_7.html#E7:33"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="matrix_7.html#E1:33_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green" title="E48">A21</font></i>: <font color="Maroon" title="c12">qq</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E1:33_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">ii</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="matrix_7.html#D4" target="_self" title="MATRIX_7:def.4">Def4</a>, <a class="txt" href="matrix_7.html#E26:33_1"><i><font color="Green" title="E47">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a>, <a class="txt" href="matrix_7.html#E11:33_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="matrix_7.html#E13:33_1"><i><font color="Green" title="E40">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">q</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:33"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="matrix_7.html#E15:33_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="matrix_7.html#E25:33_1"><i><font color="Green" title="E46">A19</font></i></a>, <a class="ref" href="matrix_7.html#T27" target="_self" title="MATRIX_7:th.27">Th27</a></i>
;<br/>
<a NAME="E28:33_1"/><i><font color="Green" title="E49">A22</font></i>: 
for <font color="Olive" title="b1">k</font> being   <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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j1</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">i1</font> &amp; <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> holds <br/><font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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" title="c17">k</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green" title="E50">A23</font></i>: 
( <font color="Maroon" title="c17">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">j1</font> &amp; <font color="Maroon" title="c17">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">i1</font> &amp; <font color="Maroon" title="c17">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> )
 ;<br/>

<a NAME="E2:33_1_2"/><b>then </b>
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E24:33_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="matrix_7.html#E22:33_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="matrix_7.html#E19:33_1"><i><font color="Green" title="E42">A15</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_2"/>
<font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E27:33_1"><i><font color="Green" title="E48">A21</font></i></a>, <a class="txt" href="matrix_7.html#E1:33_1_2"><i><font color="Green" title="E50">A23</font></i></a>, <a class="txt" href="matrix_7.html#E22:33_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="matrix_7.html#E24:33_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:33_1"/><i><font color="Green" title="E50">A24</font></i>: 
<font color="Maroon" title="c3">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">l</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a>, <a class="txt" href="matrix_7.html#E13:33_1"><i><font color="Green" title="E40">A13</font></i></a></i>;<br/>
<i><font color="Green" title="E51">A25</font></i>: <font color="Maroon" title="c12">qq</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="matrix_7.html#K5" title="MATRIX_7:func.5">"</a> </span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">i</font>
<b>by </b><i><a class="txt" href="matrix_7.html#E1:33_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">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" title="FINSEQ_5:func.4">Rev</a> <font color="Maroon" title="c3">l</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c10">ii</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="matrix_7.html#D4" target="_self" title="MATRIX_7:def.4">Def4</a>, <a class="txt" href="matrix_7.html#E26:33_1"><i><font color="Green" title="E47">A20</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">l</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c9">j</font></span>)</span> <a href="group_1.html#K3" title="GROUP_1:func.3">"</a> 
<b>by </b><i><a class="ref" href="finseq_5.html#T69" title="FINSEQ_5:th.69">FINSEQ_5:69</a>, <a class="txt" href="matrix_7.html#E11:33_1"><i><font color="Green" title="E39">A12</font></i></a>, <a class="txt" href="matrix_7.html#E13:33_1"><i><font color="Green" title="E40">A13</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">q</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a> 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:33"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="matrix_7.html#E15:33_1"><i><font color="Green" title="E41">A14</font></i></a>, <a class="txt" href="matrix_7.html#E29:33_1"><i><font color="Green" title="E50">A24</font></i></a>, <a class="ref" href="matrix_7.html#T27" target="_self" title="MATRIX_7:th.27">Th27</a></i>
;<br/>
<a NAME="E31:33_1"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> &amp; <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">i</font> &amp; ( for <font color="Olive" title="b3">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> holds <br/><font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">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" title="c17">i</font> = <font color="Maroon" title="c13">i1</font>;<br/>

<b>take </b><font color="Maroon" title="c18">j</font> = <font color="Maroon" title="c14">j1</font>;<br/>

<b>thus </b><a NAME="E1:33_1_4"/>
( <font color="Maroon" title="c17">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> &amp; <font color="Maroon" title="c18">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> &amp; <font color="Maroon" title="c17">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">j</font> &amp; <font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j</font> &amp; <font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i</font> )
 <b>by </b><i><a class="txt" href="matrix_7.html#E19:33_1"><i><font color="Green" title="E42">A15</font></i></a>, <a class="txt" href="matrix_7.html#E22:33_1"><i><font color="Green" title="E44">A17</font></i></a>, <a class="txt" href="matrix_7.html#E24:33_1"><i><font color="Green" title="E45">A18</font></i></a>, <a class="txt" href="matrix_7.html#E30:33_1"><i><font color="Green" title="E51">A25</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>

<b>let </b><font color="Maroon" title="c19">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<a NAME="E2:33_1_4"/>
<font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_4"/>
( <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c17">i</font> &amp; <font color="Maroon" title="c19">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c18">j</font> &amp; <font color="Maroon" title="c19">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c12">qq</font> implies <font color="Maroon" title="c12">qq</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">k</font> )
 <b>by </b><i><a class="txt" href="matrix_7.html#E28:33_1"><i><font color="Green" title="E49">A22</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:33_1"/><b>then </b>
<font color="Maroon" title="c12">qq</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a> 
 
<b>by </b><i><a class="ref" href="matrix_2.html#D14" title="MATRIX_2:def.14">MATRIX_2:def 14</a></i>;<br/>
<b>hence </b><a NAME="E33:33_1"/>
 ex <font color="Olive" title="b1">q2</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="Maroon" title="c4">lv</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q2</font> &amp; <font color="Olive" title="b1">q2</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:33"/>
<font color="Maroon" title="c2">IT</font> <a href="matrix_7.html#K3" title="MATRIX_7:func.3">"</a>  is <a href="matrix_2.html#V3" title="MATRIX_2:attr.3">even</a>
 <b>by </b><i><a class="ref" href="matrix_2.html#D15" title="MATRIX_2:def.15">MATRIX_2:def 15</a>, <a class="txt" href="matrix_7.html#E2:33"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="matrix_7.html#E4:33"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="matrix_7.html#E7:33"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="matrix_7.html#E11:33"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>


</div>
