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



<b>assume </b><a NAME="E1:51"/><i><font color="Green">E31</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; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">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>  ) ) )
 ;<br/>

<a NAME="E2:51"/>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</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="E3:51"/><b>then </b>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <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/>
<a NAME="E4:51"/><b>then </b>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <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/>
<b>then reconsider </b><font color="Maroon">Pf</font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E6:51"/>
 ex <font color="Olive">l</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</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> st <br/>( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">l</font></span>)</span> <a href="nat_1.html#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">Pf</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Olive">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="setwop_2.html#K3">dom</a> <font color="Olive">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="Olive">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>  ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">l</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</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="E2:51_1"/><i><font color="Green">E34</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">Pf</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> )
 <b>and </b><br/><a NAME="E3:51_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">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/>;<br/>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<b>thus </b><a NAME="E4:51_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#K6">mod</a> 2 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">Pf</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> )
 <b>by </b><i/>;<br/>

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

<a NAME="E5:51_1"/>
<font color="Maroon">i</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="E6:51_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">l</font> implies  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="Maroon">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/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:51"/><b>then </b>
<font color="Maroon">Pf</font> 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></i>;<br/>
<b>hence </b><a NAME="E8:51"/>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> is  <a href="matrix_2.html#V3">even</a> <a href="funct_2.html#NM2">Permutation</a> of  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/>


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