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

<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>let </b><font color="Maroon" title="c2">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E38">A1</font></i>: 
<font color="Maroon" title="c1">i</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="c2">M</font></span>)</span>
 ;<br/>

<a NAME="E2:51"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<i><font color="Green" title="E40">A3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
;<br/>
<a NAME="E4:51"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:51_2"/><i><font color="Green" title="E41">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> )
 ;<br/>

<a NAME="E2:51_2"/><i><font color="Green" title="E42">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:51_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E3:51_2"/><i><font color="Green" title="E43">A6</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:51_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E3:51"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>
<a NAME="E4:51_2"/><i><font color="Green" title="E44">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:51_2"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E5:51_2"/><i><font color="Green" title="E45">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i</font> &amp; <font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:51"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>
<a NAME="E6:51_2"/><i><font color="Green" title="E46">A9</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">M</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E5:51_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="matrixc1.html#E2:51_2"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<a NAME="E7:51_2"/><i><font color="Green" title="E47">A10</font></i>: 
<font color="Maroon" title="c3">j</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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:51_2"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E2:51"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/>
<a NAME="E8:51_2"/><i><font color="Green" title="E48">A11</font></i>: 
<font color="Maroon" title="c3">j</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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E7:51_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E7:51_2"><i><font color="Green" title="E47">A10</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a>, <a class="txt" href="matrixc1.html#E6:51_2"><i><font color="Green" title="E46">A9</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E8:51_2"><i><font color="Green" title="E48">A11</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font>
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a>, <a class="txt" href="matrixc1.html#E4:51_2"><i><font color="Green" title="E44">A7</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E10:51_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:51"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">M</font>,<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<font color="Maroon" title="c1">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E2:51"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="matrixc1.html#E3:51"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>


</div>
