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

<b>let </b><font color="Maroon">n</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="vectsp_1.html#NM2">Field</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">K</font>;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>, <font color="Maroon">N</font> be   <a href="matrix_1.html#NM5">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">K</font>;<br/>







<b>assume </b><a NAME="E1:71"/><i><font color="Green">E31</font></i>: 
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Olive">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">i</font> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">l</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Olive">l</font> ) )
 ;<br/>

<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:71"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:71"/><i><font color="Green">E37</font></i>: 
( ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</font></span>)</span> ) &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">i</font> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> holds <br/> <a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">l</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Olive">l</font> ) )
 <b>by </b><i><a class="ref" href="matrix_9.html#T3" target="_self">Th3</a></i>;<br/>
<b>let </b><font color="Maroon">p</font> be    <a href="matrix_2.html#M3">Element</a> of  <a href="matrix_2.html#K12">Permutations</a> <font color="Maroon">n</font>;<br/>

<a NAME="E5:71"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E6:71"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:71"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E9:71"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E10:71"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="matrix_9.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E11:71"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<a NAME="E12:71"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, </i>;<br/>
<a NAME="E13:71"/><b>then </b><i><font color="Green">E42</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">N</font> )
 
<b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E14:71"/><b>then </b><i><font color="Green">E43</font></i>: 
( <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> )
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E15:71"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><b>proof </b></a><div class="add">

<a NAME="E1:71_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E2:71_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:71_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<a NAME="E4:71_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E5:71_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/>
<a NAME="E6:71_1"/><b>then </b>
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E7:71_1"/><b>then </b>
<font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 
<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/>
<b>hence </b><a NAME="E8:71_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">N</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E16:71"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">M</font></span>)</span> &amp; <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="matrix_9.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E17:71"/><b>then </b>
( <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">k</font>,<font color="Maroon">i</font> )
 
<b>by </b><i>, <a class="ref" href="matrix_3.html#D7">MATRIX_3:def 7</a></i>;<br/>
<b>then </b><i><font color="Green">E73</font></i>: <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">M</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>
<b>by </b><i><a class="ref" href="matrix_9.html#D1" target="_self">Def1</a>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="matrix_9.html#T5" target="_self">Th5</a></i>
;<br/>
<b>take </b>
<font color="Maroon">k</font>
;<br/>

<b>thus </b><a NAME="E19:71"/>
( <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">M</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_3.html#K10">Path_matrix</a> <font color="Maroon">p</font>,<font color="Maroon">N</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="matrix_9.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a>, </i>;<br/>


</div>
