<?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="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M1</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">n</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">M2</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">m</font>,<font color="Maroon">k</font>,<font color="Maroon">D</font>;<br/>











<b>assume </b><a NAME="E1:43"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font>
 ;<br/>

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

<b>assume </b><a NAME="E2:43"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span>
 ;<br/>

<i><font color="Green">E50</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span>
 ;<br/><a NAME="E2:43_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></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="E3:43_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E4:43_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:43_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E6:43_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p3">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:43_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">P</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E9:43_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<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">i</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_2_1_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> ) )
 </span><b>by </b><i>, <a class="ref" href="finseq_1.html#T38">FINSEQ_1:38</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_1"><b>suppose </b></a><a NAME="E1:43_2_1_1_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:43_2_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></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="E3:43_2_1_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M1</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E4:43_2_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:43_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M1</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M1</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E6:43_2_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">P1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E8:43_2_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> &amp; <font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/><a NAME="E9:43_2_1_1_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</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/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_2_1_1_2"><b>suppose </b></a><a NAME="E1:43_2_1_1_2"/>
 ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Olive">n</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">n</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="E3:43_2_1_1_2"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font> )
 ;<br/><a NAME="E4:43_2_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></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="E5:43_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M2</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E6:43_2_1_1_2"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:43_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M2</font></span>)</span></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i>, , <a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:43_2_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">n</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D5">MATRIX_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">P2</font> being    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E10:43_2_1_1_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">P2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/><a NAME="E11:43_2_1_1_2"/><i><font color="Green">E111</font></i>: 
 <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">M1</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">M1</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E12:43_2_1_1_2"/>
 <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">M2</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">M2</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D9">MATRIX_1:def 9</a></i>;<br/><a NAME="E13:43_2_1_1_2"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/><a NAME="E14:43_2_1_1_2"/><b>then </b>
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> = 
<font color="Maroon">M2</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">n</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/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E11:43_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:43"/>
 <a href="matrix_1.html#K9">Col</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="matrlin.html#K10" target="_self">^</a> <font color="Maroon">M2</font></span>)</span>,<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">M1</font>,<font color="Maroon">i</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M2</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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