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

<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>;<br/>



<b>set </b><font color="Maroon" title="c3">n2</font> = <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2;<br/>
<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">G</font> =  <a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><br/> for <font color="Olive" title="b2">p2</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>  st <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">P</font> holds <br/> ex <font color="Olive" title="b4">trans</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">trans</font> &amp; <font color="Olive" title="b4">trans</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  ) ) holds <br/>( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> ) &amp; ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) );<br/>
<a NAME="E1:23"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">P</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">p2</font> be    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:23_1"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c7">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>and </b><br/><a NAME="E2:23_1"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">P</font> holds <br/> ex <font color="Olive" title="b2">trans</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c6">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">trans</font> &amp; <font color="Olive" title="b2">trans</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 ;<br/>

<a NAME="E3:23_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 
<b>by </b><i><a class="txt" href="matrix11.html#E1:23_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:23_1"/><b>then </b>
<font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></i>;<br/>
<a NAME="E5:23_1"/><b>then </b>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_4.html#T11" title="GROUP_4:th.11">GROUP_4:11</a></i>;<br/>
<a NAME="E6:23_1"/><b>then </b>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c6">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T28" title="MATRIX_2:th.28">MATRIX_2:28</a></i>;<br/>
<b>hence </b><a NAME="E7:23_1"/>
( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> ) &amp; ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c7">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:23_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="matrix11.html#T12" target="_self" title="MATRIX11:th.12">Th12</a>, <a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:23"/><i><font color="Green" title="E20">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:23_2"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>]
 ;<br/>

<b>set </b><font color="Maroon" title="c7">k1</font> = <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/>
<b>let </b><font color="Maroon" title="c8">P</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span>;<br/><b>let </b><font color="Maroon" title="c9">p2</font> be    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:23_2"/><i><font color="Green" title="E22">A5</font></i>: 
( <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c8">P</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>and </b><br/><a NAME="E3:23_2"/><i><font color="Green" title="E23">A6</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P</font> holds <br/> ex <font color="Olive" title="b2">trans</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c8">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">trans</font> &amp; <font color="Olive" title="b2">trans</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 ;<br/>

<a NAME="E4:23_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E2:23_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c11">Q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E6:23_2"/><i><font color="Green" title="E24">A7</font></i>: 
( <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c11">Q</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">Q</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a>, <a class="ref" href="rewrite1.html#T5" title="REWRITE1:th.5">REWRITE1:5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">X</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>, <font color="Maroon" title="c13">Q</font> = <font color="Maroon" title="c11">Q</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <b>by </b><i><a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="finseq_1.html#T50" title="FINSEQ_1:th.50">FINSEQ_1:50</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">PQ</font> =  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c13">Q</font> as    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>;<br/>
<a NAME="E9:23_2"/><i><font color="Green" title="E25">A8</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">Q</font> holds <br/> ex <font color="Olive" title="b2">trans</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c13">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">trans</font> &amp; <font color="Olive" title="b2">trans</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:23_2_1"/><i><font color="Green" title="E26">A9</font></i>: 
<font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">Q</font>
 ;<br/>

<a NAME="E2:23_2_1"/>
<font color="Maroon" title="c15">i</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:23_2_1"/><b>then </b>
( <font color="Maroon" title="c13">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">X</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">i</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">X</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="matrix11.html#E1:23_2_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="ref" href="finseq_1.html#T41" title="FINSEQ_1:th.41">FINSEQ_1:41</a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:23_2_1"/>
 ex <font color="Olive" title="b1">trans</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Maroon" title="c13">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">trans</font> &amp; <font color="Olive" title="b1">trans</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:23_2"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:23_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E11:23_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E12:23_2"/><b>then </b>
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">P</font> &amp; <font color="Maroon" title="c8">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">x</font> )
 
<b>by </b><i><a class="txt" href="matrix11.html#E2:23_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">tr</font> being    <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span><b> such that </b><br/><a NAME="E14:23_2"/><i><font color="Green" title="E26">A10</font></i>: 
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">tr</font> &amp; <font color="Maroon" title="c15">tr</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  )
 <b>by </b><i><a class="txt" href="matrix11.html#E3:23_2"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">Tr</font> = <font color="Maroon" title="c15">tr</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <b>by </b><i><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>;<br/>
<i><font color="Green" title="E27">A11</font></i>: <font color="Maroon" title="c9">p2</font> = 
<font color="Maroon" title="c16">Tr</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c13">Q</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix11.html#E2:23_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="matrix11.html#E14:23_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="group_4.html#T10" title="GROUP_4:th.10">GROUP_4:10</a></i>
<br/>.= 
the <a href="group_1.html#U1" title="GROUP_1:sel.1">mult</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">Tr</font>,<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Maroon" title="c13">Q</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D1" title="GROUP_1:def.1">GROUP_1:def 1</a></i>
<br/>.= 
<font color="Maroon" title="c14">PQ</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">tr</font>
<b>by </b><i><a class="ref" href="matrix_2.html#D13" title="MATRIX_2:def.13">MATRIX_2:def 13</a></i>
;<br/>
<a NAME="E17:23_2"/><b>then </b><i><font color="Green" title="E28">A12</font></i>: 
 <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c14">PQ</font>,<font color="Maroon" title="c2">K</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrix11.html#E14:23_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="matrix11.html#T13" target="_self" title="MATRIX11:th.13">Th13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_2_3_1"/>
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">Q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">Q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_d.html#T12" title="NAT_D:th.12">NAT_D:12</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_3_1_1"><b>suppose </b></a><a NAME="E1:23_2_3_1_1"/><i><font color="Green" title="E29">A13</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">Q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ;<br/><div class="add"><a NAME="E2:23_2_3_1_1"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c14">PQ</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> &amp;  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:23_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="matrix11.html#E2:23_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="matrix11.html#E9:23_2"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/><a NAME="E3:23_2_3_1_1"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="matrix11.html#E14:23_2"><i><font color="Green" title="E26">A10</font></i></a>, <a class="txt" href="matrix11.html#E16:23_2"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="matrix11.html#E1:23_2_3_1_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="matrix11.html#T13" target="_self" title="MATRIX11:th.13">Th13</a>, <a class="ref" href="radix_1.html#T3" title="RADIX_1:th.3">RADIX_1:3</a></i>;<br/><b>hence </b><a NAME="E4:23_2_3_1_1"/>
( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> ) &amp; ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_2_3_1_2"><b>suppose </b></a><a NAME="E1:23_2_3_1_2"/><i><font color="Green" title="E29">A14</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">Q</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:23_2_3_1_2"/><b>then </b>
(  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c14">PQ</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> &amp; 2 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="matrix11.html#E1:23_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="matrix11.html#E2:23_2"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="matrix11.html#E9:23_2"><i><font color="Green" title="E25">A8</font></i></a></i>;<br/><b>hence </b><a NAME="E3:23_2_3_1_2"/>
( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> ) &amp; ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="matrix11.html#E6:23_2"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="matrix11.html#E17:23_2"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="matrix11.html#E1:23_2_3_1_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="radix_1.html#T2" title="RADIX_1:th.2">RADIX_1:2</a>, <a class="ref" href="rlvect_1.html#T30" title="RLVECT_1:th.30">RLVECT_1:30</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:23_2"/>
( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> ) &amp; ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Maroon" title="c9">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:23"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="matrix11.html#E1:23"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="matrix11.html#E2:23"><i><font color="Green" title="E20">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:23"/>
for <font color="Olive" title="b1">P</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="matrix_2.html#K13" title="MATRIX_2:func.13">Group_of_Perm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span></span>)</span><br/> for <font color="Olive" title="b2">p2</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span>  st <font color="Olive" title="b2">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <font color="Olive" title="b1">P</font> &amp; ( for <font color="Olive" title="b3">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Olive" title="b1">P</font> holds <br/> ex <font color="Olive" title="b4">trans</font> being   <a href="matrix_2.html#M3" title="MATRIX_2:mode.3">Element</a> of  <a href="matrix_2.html#K12" title="MATRIX_2:func.12">Permutations</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">trans</font> &amp; <font color="Olive" title="b4">trans</font> <a href="matrix_2.html#NR1" title="MATRIX_2:NR.1">is_transposition</a>  ) ) holds <br/>( ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font> ) &amp; ( <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">P</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="matrix11.html#K2" title="MATRIX11:func.2">sgn</a> <font color="Olive" title="b2">p2</font>,<font color="Maroon" title="c2">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c2">K</font></span>)</span> ) )
 ;<br/>


</div>
