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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:34"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="matrix_2.html#V3">even</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<a NAME="E2:34"/><i><font color="Green">E31</font></i>: 
 <a href="matrix_2.html#K10">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font> being   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E4:34"/><i><font color="Green">E32</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>3</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></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>, <a class="txt" href="matrix_7.html#E1:34"><i><font color="Green">E30</font></i></a>, <a class="txt" href="matrix_7.html#E2:34"><i><font color="Green">E31</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a>  as   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> ;<br/>
<a NAME="E6:34"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E7:34"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="matrix_7.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E8:34"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></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">c<sub>5</sub></font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E11:34"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="group_1.html#K3">"</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="matrix_7.html#K3" target="_self">"</a> 
 
<b>by </b><i><a class="ref" href="matrix_7.html#T27" target="_self">Th27</a>, <a class="txt" href="matrix_7.html#E1:34"><i><font color="Green">E30</font></i></a></i>;<br/>
<i><font color="Green">E37</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="matrix_7.html#K3" target="_self">"</a>  = 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrix_7.html#E11:34"><i><font color="Green">E36</font></i></a></i>
<br/>.= 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="matrix_7.html#T26" target="_self">Th26</a></i>
;<br/>
<a NAME="E13:34"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:34_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:34_2"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E7:34"><i><font color="Green">E34</font></i></a>, <a class="txt" href="matrix_7.html#E2:34_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E4:34_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></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="E5:34_2"/><b>then </b><i><font color="Green">E42</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E6:34_2"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a>, <a class="txt" href="matrix_7.html#E3:34_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E7:34_2"/><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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E8:34_2"/><b>then </b><i><font color="Green">E43</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">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E9:34_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K9">+</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">c<sub>3</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i><a class="txt" href="matrix_7.html#E4:34_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K10">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="binop_2.html#K9">+</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<a NAME="E11:34_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="binop_2.html#K23">+</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> 1
 
;<br/>
<a NAME="E12:34_2"/>
<font color="Maroon">c<sub>9</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E5:34_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="matrix_7.html#E8:34_2"><i><font color="Green">E43</font></i></a></i>;<br/>
<a NAME="E13:34_2"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>3</sub></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">c<sub>10</sub></font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E15:34_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix_7.html#E4:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrix_7.html#E13:34_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E16:34_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></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#E1:34_2"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></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">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E19:34_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="matrix_7.html#E15:34_2"><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">c<sub>14</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E21:34_2"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></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>, <a class="txt" href="matrix_7.html#E1:34"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E22:34_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></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">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>, <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E24:34_2"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="txt" href="matrix_7.html#E21:34_2"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E25:34_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> )
 
<b>by </b><i><a class="txt" href="matrix_7.html#E19:34_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="matrix_7.html#E22:34_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrix_7.html#E24:34_2"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E26:34_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></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#E13:34_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E27:34_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E15:34_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="matrix_7.html#E26:34_2"><i><font color="Green">E52</font></i></a></i>;<br/>
<a NAME="E28:34_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>8</sub></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">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E1:34_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="matrix_7.html#E8:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrix_7.html#E6:34"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<i><font color="Green">E55</font></i>: <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrix_7.html#E1:34_2"><i><font color="Green">E38</font></i></a>, <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">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="matrix_7.html#D4" target="_self">Def4</a>, <a class="txt" href="matrix_7.html#E28:34_2"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></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#E11:34_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_7.html#E13:34_2"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="matrix_7.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="matrix_7.html#E27:34_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="matrix_7.html#T27" target="_self">Th27</a>, <a class="txt" href="matrix_7.html#E1:34"><i><font color="Green">E30</font></i></a></i>
;<br/>
<a NAME="E30:34_2"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> holds <br/><font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:34_2_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<br/>

<a NAME="E2:34_2_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E24:34_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="matrix_7.html#E22:34_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrix_7.html#E19:34_2"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:34_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="matrix_7.html#E29:34_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrix_7.html#E1:34_2_2"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrix_7.html#E22:34_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrix_7.html#E24:34_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:34_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></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#E13:34_2"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E32:34_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E15:34_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="matrix_7.html#E31:34_2"><i><font color="Green">E57</font></i></a></i>;<br/>
<i><font color="Green">E59</font></i>: <font color="Maroon">c<sub>11</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_5.html#K4">Rev</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="matrix_7.html#K5" target="_self">"</a> </span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="matrix_7.html#E1:34_2"><i><font color="Green">E38</font></i></a>, <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">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="group_1.html#K3">"</a> 
<b>by </b><i><a class="ref" href="matrix_7.html#D4" target="_self">Def4</a>, <a class="txt" href="matrix_7.html#E28:34_2"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></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#E11:34_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="matrix_7.html#E13:34_2"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>10</sub></font> <a href="matrix_7.html#K3" target="_self">"</a> 
<b>by </b><i><a class="txt" href="matrix_7.html#E32:34_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrix_7.html#T27" target="_self">Th27</a>, <a class="txt" href="matrix_7.html#E1:34"><i><font color="Green">E30</font></i></a></i>
;<br/>
<a NAME="E34:34_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E19:34_2"><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#E33:34_2"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E35:34_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="matrix_7.html#E19:34_2"><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#E33:34_2"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E36:34_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></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>, <a class="txt" href="matrix_7.html#E25:34_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrix_7.html#E19:34_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="matrix_7.html#E30:34_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="matrix_7.html#E34:34_2"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E37:34_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K11">Permutations</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:34"/>
<font color="Maroon">c<sub>2</sub></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#E2:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="matrix_7.html#E4:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="matrix_7.html#E7:34"><i><font color="Green">E34</font></i></a>, <a class="txt" href="matrix_7.html#E12:34"><i><font color="Green">E37</font></i></a></i>;<br/>


</div>
