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

<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> 3</span>)</span>;<br/>

<b>assume </b><a NAME="E1:54"/><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> 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/>

<b>then consider </b><font color="Maroon">t</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54"/><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> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 )
 <b>by </b><i><a class="ref" href="nat_1.html#D2">NAT_1:def 2</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">t</font>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E4"><i><font color="Green">Lemma47</font></i></a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] <b>means </b>for <font color="Olive">f</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  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">a<sub>1</sub></font> &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="Olive">f</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">f</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>  ) ) &amp; not  <a href="group_4.html#K3">Product</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; not  <a href="group_4.html#K3">Product</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <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> holds <br/> <a href="group_4.html#K3">Product</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span>;<br/>
<a NAME="E5:54"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</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> 3</span>)</span>;<br/>

<b>assume </b><a NAME="E1:54_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</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">f</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">f</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="E2:54_1"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="matrix_2.html#K13">Group_of_Perm</a> 3;<br/>
<a NAME="E3:54_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <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">Group_of_Perm</a> 3</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/>
<a NAME="E4:54_1"/>
 <a href="group_1.html#K2">1.</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13">Group_of_Perm</a> 3</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T28">MATRIX_2:28</a>, <a class="ref" href="finseq_2.html#T62">FINSEQ_2:62</a></i>;<br/>
<b>hence </b><a NAME="E5:54_1"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="group_4.html#T11">GROUP_4:11</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:54"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="xcmplx_0.html#K2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:54_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:54_2"/><b>then </b><i><font color="Green">E42</font></i>: 
for <font color="Olive">f</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  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">k</font> &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="Olive">f</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">f</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>  ) ) &amp; not  <a href="group_4.html#K3">Product</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; not  <a href="group_4.html#K3">Product</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <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> holds <br/> <a href="group_4.html#K3">Product</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span>
 
;<br/>
<b>let </b><font color="Maroon">f</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> 3</span>)</span>;<br/>

<b>assume </b><a NAME="E3:54_2"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> 2 <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &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">f</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">f</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/>

<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:54_2"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</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> 3</span>)</span>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E6:54_2"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T99">RELAT_1:99</a></i>;<br/>
<a NAME="E7:54_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</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> 3</span>)</span>
 
<b>by </b><i><a class="txt" href="matrix_9.html#E6"><i><font color="Green">Lemma53</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</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> <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E9:54_2"/>
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:54_2"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<b>set </b><font color="Maroon">l</font> = 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>;<br/>
<a NAME="E11:54_2"/><i><font color="Green">E72</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E12:54_2"/><b>then </b>
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E13:54_2"/><b>then </b>
 <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E14:54_2"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> 2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<a NAME="E15:54_2"/><i><font color="Green">E74</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T89">RELAT_1:89</a></i>;<br/>
<a NAME="E16:54_2"/>
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">g</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">g</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="54_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:54_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> 3<b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i><a class="ref" href="matrix_9.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>take </b>
<font color="Maroon">q</font>
;<br/>

<a NAME="E4:54_2_1"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="matrix_9.html#T6" target="_self">Th6</a>, <a class="ref" href="matrix_9.html#T5" target="_self">Th5</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<b>hence </b><a NAME="E5:54_2_1"/>
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">q</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i><a class="ref" href="matrix_9.html#T6" target="_self">Th6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:54_2"/><b>then </b><i><font color="Green">E78</font></i>: 
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i>, <a class="ref" href="matrix_9.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E18:54_2"/><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E19:54_2"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<b>set </b><font color="Maroon">h</font> = <font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>;<br/>
<a NAME="E20:54_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="matrix_9.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_3.html#T59">FINSEQ_3:59</a></i>;<br/>
<a NAME="E21:54_2"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p3">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p4">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>;<br/>
<a NAME="E22:54_2"/><i><font color="Green">E83</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E23:54_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E24:54_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="matrix_9.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E25:54_2"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E26:54_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> 2 <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E27:54_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="matrix_9.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E28:54_2"/><i><font color="Green">E88</font></i>: 
2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E29:54_2"/>
<span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<a NAME="E30:54_2"/><b>then </b><i><font color="Green">E141</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T72">FUNCT_1:72</a></i>;<br/>
<i><font color="Green">E142</font></i>: <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>

<br/>.= 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_5.html#T80">FINSEQ_5:80</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span></span>)</span>

;<br/>
<a NAME="E32:54_2"/><i><font color="Green">E143</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><a href="setwop_2.html#K2">Seg</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrix_9.html#D1" target="_self">Def1</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E33:54_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E34:54_2"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="binarith.html#K10">^</a> <span class="p2"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span></span>)</span>
 
<b>by </b><i>, , , , <a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<b>then </b><i><font color="Green">E144</font></i>:  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span> <a href="binarith.html#K10">^</a> <span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8">GROUP_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p3"><a href="matrix_2.html#K4">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="matrix_2.html#K4">*&gt;</a></span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T9">GROUP_4:9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T12">GROUP_4:12</a></i>
;<br/>
<b>consider </b><font color="Maroon">q2</font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> 3<b> such that </b><br/><a NAME="E37:54_2"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i>, <a class="ref" href="matrix_9.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E38:54_2"/><i><font color="Green">E146</font></i>: 
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E39:54_2"/><i><font color="Green">E147</font></i>: 
 <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/>
<b>consider </b><font color="Maroon">q1</font> being    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> 3<b> such that </b><br/><a NAME="E41:54_2"/><i><font color="Green">E148</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">q1</font> <a href="matrix_2.html#NR1" target="_self">is_transposition</a>  )
 <b>by </b><i>, </i>;<br/>
<a NAME="E42:54_2"/><i><font color="Green">E149</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span>
 
<b>by </b><i>, <a class="txt" href="matrix_9.html#E15:54_2"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E43:54_2"/><i><font color="Green">E150</font></i>: 
( <font color="Maroon">q1</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> or <font color="Maroon">q1</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> or <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,2,1</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="matrix_9.html#E14:54_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="matrix_9.html#E15:54_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="matrix_9.html#E5:54"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E44:54_2"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">q2</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> or <font color="Maroon">q2</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> or <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,2,1</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i>, <a class="txt" href="matrix_9.html#E14:54_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="matrix_9.html#E5:54"><i><font color="Green">E38</font></i></a></i>;<br/>
<a NAME="E45:54_2"/><i><font color="Green">E152</font></i>: 
<font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R2">in</a>  <a href="matrix_2.html#K12">Permutations</a> 3
 
<b>by </b><i/>;<br/>
<a NAME="E46:54_2"/><b>then </b><i><font color="Green">E153</font></i>: 
<font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> is   <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></i>;<br/>
<a NAME="E47:54_2"/>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">g</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> 3</span>)</span>
 
;<br/>
<a NAME="E48:54_2"/><b>then </b>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">g</font> <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_2.html#D13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E49:54_2"/><b>then </b><i><font color="Green">E154</font></i>: 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">g</font> is   <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></i>;<br/>
<b>then reconsider </b><font color="Maroon">Pg</font> =  <a href="group_4.html#K3">Product</a> <font color="Maroon">g</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_2.html#D11">MATRIX_2:def 11</a></i>;<br/>
<a NAME="E51:54_2"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <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> or <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="matrix_9.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E52:54_2"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">q1</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> or <font color="Maroon">q1</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> or <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,2,1</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E53:54_2"/>
( <font color="Maroon">q2</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> or <font color="Maroon">q2</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> or <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,2,1</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E54:54_2"/><b>then </b><i><font color="Green">E157</font></i>: 
( <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <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> or <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="matrix_9.html#E21:54_2"><i><font color="Green">E82</font></i></a>, <a class="txt" href="matrix_9.html#E4:54"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E55:54_2"/><i><font color="Green">E158</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font>
 
<b>by </b><i>, <a class="txt" href="matrix_9.html#E11:54_2"><i><font color="Green">E72</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">q12</font> = <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</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="txt" href="matrix_9.html#E17:54_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">f12</font> = <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="nat_1.html#K2">*</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span></span>)</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> ;<br/>
<a NAME="E58:54_2"/>
 <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">g</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">f12</font>
 
<b>by </b><i/>;<br/>
<a NAME="E59:54_2"/><b>then </b><i><font color="Green">E159</font></i>: 
 <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q12</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">Pg</font>
 
<b>by </b><i>, <a class="txt" href="matrix_9.html#E24:54_2"><i><font color="Green">E84</font></i></a></i>;<br/>
<a NAME="E60:54_2"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_4"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_2_4_1"/>
( ( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <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> ) or ( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <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> ) or ( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> ) or <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or ( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> ) or ( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <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> ) or ( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> ) )
 </span><b>by </b><i><a class="txt" href="matrix_9.html#E19:54_2"><i><font color="Green">E81</font></i></a>, <a class="txt" href="matrix_9.html#E22:54_2"><i><font color="Green">E83</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1_1"><b>suppose </b></a><a NAME="E1:54_2_4_1_1"/>
( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_4_1_1"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="matrix_9.html#E3:54_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_9.html#E18:54_2"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1_2"><b>suppose </b></a><a NAME="E1:54_2_4_1_2"/>
( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_4_1_2"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E4:54"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1_3"><b>suppose </b></a><a NAME="E1:54_2_4_1_3"/>
( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <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> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_4_1_3"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E4:54"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1_4"><b>suppose </b></a><a NAME="E1:54_2_4_1_4"/>
<font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_4_1_4"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E19:54_2"><i><font color="Green">E81</font></i></a>, , , <a class="txt" href="matrix_9.html#E5:54_2"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1_5"><b>suppose </b></a><a NAME="E1:54_2_4_1_5"/>
( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_4_1_5"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i>, <a class="txt" href="matrix_9.html#E3:54_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrix_9.html#E18:54_2"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1_6"><b>suppose </b></a><a NAME="E1:54_2_4_1_6"/>
( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_4_1_6"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E4:54"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_2_4_1_7"><b>suppose </b></a><a NAME="E1:54_2_4_1_7"/>
( <font color="Maroon">Pg</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> &amp; <font color="Maroon">q1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_2_4_1_7"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="matrix_9.html#E4:54"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E61:54_2"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:54"/>
for <font color="Olive">s</font> being  <a href="nat_1.html#NM1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S2">NAT_1:sch 2</a>(, );<br/></i>
<b>hence </b><a NAME="E8:54"/>
(  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">1,2,3</span><a href="finseq_4.html#K3">*&gt;</a></span> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <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> or  <a href="group_4.html#K3">Product</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K3">&lt;*</a><span class="default">3,1,2</span><a href="finseq_4.html#K3">*&gt;</a></span> )
 <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#E5"><i><font color="Green">Lemma48</font></i></a></i>;<br/>


</div>
