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

<b>reconsider </b><font color="Maroon">f</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3">*&gt;</a></span> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="setwop_2.html#K2">Seg</a> 3 <b>by </b><i><a class="ref" href="matrix_2.html#D11">MATRIX_2:def 11</a>, <a class="ref" href="matrix_9.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:49"/>
 ex <font color="Olive">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> 3</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">f</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> 3 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="49_1"><b>proof </b></a><div class="add">

<a NAME="E1:49_1"/><i><font color="Green">E31</font></i>: 
( <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,1,3</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> 3 &amp; <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> 3 )
 
<b>by </b><i><a class="ref" href="matrix_9.html#T1" target="_self">Th1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l1</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,1,3</span><a href="finseq_4.html#K3">*&gt;</a></span>, <font color="Maroon">l2</font> = <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3">*&gt;</a></span> 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> 3</span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">l</font> = <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><font color="Maroon">l2</font>,<font color="Maroon">l1</font></span><a href="finseq_4.html#K2">*&gt;</a></span> 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> 3</span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">l</font>
;<br/>

<a NAME="E4:49_1"/><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E5:49_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
 <b>by </b><i><a class="ref" href="nat_1.html#T63">NAT_1:63</a></i>;<br/>

<b>reconsider </b><font color="Maroon">L2</font> = <font color="Maroon">l2</font>, <font color="Maroon">L1</font> = <font color="Maroon">l1</font> as    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> 3 <b>by </b><i/>;<br/>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> = 
<font color="Maroon">l2</font> <a href="group_1.html#K1">*</a> <font color="Maroon">l1</font>
<b>by </b><i><a class="ref" href="group_4.html#T13">GROUP_4:13</a></i>
<br/>.= 
the <a href="group_1.html#U1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 3</span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">l2</font>,<font color="Maroon">l1</font>
<b>by </b><i><a class="ref" href="group_1.html#D1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon">L1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">L2</font>
<b>by </b><i><a class="ref" href="matrix_2.html#D13">MATRIX_2:def 13</a></i>
<br/>.= 
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3">*&gt;</a></span>
<b>by </b><i><a class="txt" href="matrix_9.html#E9:49_1"><i><font color="Green">E37</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:49_1"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font>
 ;<br/>

<a NAME="E9:49_1"/><i><font color="Green">E37</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/>
<a NAME="E10:49_1"/>
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="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> 3 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>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2"><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="E1:49_1_2"/>
<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>
 ;<br/>

<a NAME="E2:49_1_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_2_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2 or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_1_1"><b>suppose </b></a><a NAME="E1:49_1_2_1_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:49_1_2_1_1"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,1,3</span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">l</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> 3 <b>by </b><i><a class="ref" href="matrix_9.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:49_1_2_1_1"/>
 <a href="matrix_2.html#K11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> 3</span>)</span> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/><a NAME="E5:49_1_2_1_1"/><b>then </b>
<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_9.html#E3"><i><font color="Green">Lemma46</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:49_1_2_1_1"/>
 ex <font color="Olive">q</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> 3 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>  )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_1_2"><b>suppose </b></a><a NAME="E1:49_1_2_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:49_1_2_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">l</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,3,2</span><a href="finseq_4.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><b>then reconsider </b><font color="Maroon">q</font> = <font color="Maroon">l</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> 3 <b>by </b><i><a class="ref" href="matrix_9.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:49_1_2_1_2"/>
 <a href="matrix_2.html#K11">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K12">Permutations</a> 3</span>)</span> <a href="hidden.html#R1">=</a> 3
 <b>by </b><i><a class="ref" href="matrix_2.html#T20">MATRIX_2:20</a></i>;<br/><a NAME="E5:49_1_2_1_2"/><b>then </b>
<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_9.html#E3"><i><font color="Green">Lemma46</font></i></a>, <a class="txt" href="matrix_9.html#E1:49_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>hence </b><a NAME="E6:49_1_2_1_2"/>
 ex <font color="Olive">q</font> being   <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> 3 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>  )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E11:49_1"/>
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="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> 3 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/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
<span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">2,3,1</span><a href="finseq_4.html#K3">*&gt;</a></span> is  <a href="matrix_2.html#V3">even</a> <a href="funct_2.html#NM2">Permutation</a> of  <a href="setwop_2.html#K2">Seg</a> 3
 <b>by </b><i><a class="ref" href="matrix_2.html#D15">MATRIX_2:def 15</a></i>;<br/>


</div>
