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<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>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:52"/>
0 <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1"><b>proof </b></a><div class="add">

<a NAME="E1:52_1"/>
0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> 0</span>)</span> <a href="nat_1.html#K1">+</a> 0
 
;<br/>
<b>hence </b><a NAME="E2:52_1"/>
0 <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:52"/><b>then </b><i><font color="Green">E23</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p3">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K4">mod</a> 2 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E3:52"/>
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#T11">GROUP_4:11</a></i>;<br/>
<a NAME="E4:52"/><b>then </b>
 <a href="finseq_2.html#K1">idseq</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E5:52"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E6:52"/>
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> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="matrix_2.html#M3" target="_self">Element</a> of  <a href="matrix_2.html#K11" target="_self">Permutations</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p2">(<span class="default"><a href="matrix_2.html#K13" target="_self">Group_of_Perm</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <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="finseq_1.html#T26">FINSEQ_1:26</a></i>;<br/>
<b>hence </b><a NAME="E7:52"/>
 <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="matrix_2.html#V3" target="_self">even</a>
 <b>by </b><i><a class="txt" href="matrix_2.html#E2:52"><i><font color="Green">E23</font></i></a>, <a class="txt" href="matrix_2.html#E5:52"><i><font color="Green">E24</font></i></a>, <a class="ref" href="matrix_2.html#D15" target="_self">Def15</a></i>;<br/>


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