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<b>let </b><font color="Maroon" title="c1">F</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</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="E1:53"/><i><font color="Green" title="E40">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <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> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 ;<br/>

<i><font color="Green" title="E41">A2</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"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c1">F</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"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c1">F</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="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="matrixc1.html#D7" target="_self" title="MATRIXC1:def.7">Def7</a></i>
;<br/>
<a NAME="E3:53"/><i><font color="Green" title="E42">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E4:53"/><i><font color="Green" title="E43">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E5:53"/><i><font color="Green" title="E44">A5</font></i>: 
 <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> <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="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<i><font color="Green" title="E45">A6</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"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</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="matrixc1.html#D7" target="_self" title="MATRIXC1:def.7">Def7</a></i>
<br/>.= 
 <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="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</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="E1:53_3"/><i><font color="Green" title="E46">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> )
 ;<br/><a NAME="E2:53_3"/><i><font color="Green" title="E47">A8</font></i>: 
<font color="Maroon" title="c3">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>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:53_3"><i><font color="Green" title="E46">A7</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#E2:53"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/><a NAME="E3:53_3"/><i><font color="Green" title="E48">A9</font></i>: 
<font color="Maroon" title="c3">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> <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#E1:53_3"><i><font color="Green" title="E46">A7</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#E2:53"><i><font color="Green" title="E41">A2</font></i></a>, <a class="txt" href="matrixc1.html#E3:53"><i><font color="Green" title="E42">A3</font></i></a></i>;<br/><a NAME="E4:53_3"/><i><font color="Green" title="E49">A10</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> <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="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2: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>, <a class="txt" href="matrixc1.html#E5:53"><i><font color="Green" title="E44">A5</font></i></a>, <a class="txt" href="matrixc1.html#E1:53"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="matrixc1.html#E4:53"><i><font color="Green" title="E43">A4</font></i></a></i>;<br/><a NAME="E5:53_3"/><i><font color="Green" title="E50">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <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="c3">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="matrixc1.html#E27"><i><font color="Green" title="E17">Lm2</font></i></a>, <a class="txt" href="matrixc1.html#E4:53_3"><i><font color="Green" title="E49">A10</font></i></a>, <a class="txt" href="matrixc1.html#E4:53"><i><font color="Green" title="E43">A4</font></i></a>, <a class="txt" href="matrixc1.html#E1:53"><i><font color="Green" title="E40">A1</font></i></a></i>;<br/><a NAME="E6:53_3"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c1">F</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:53_3"><i><font color="Green" title="E46">A7</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c1">F</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">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">F</font>,<span class="p3">(<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="c3">i</font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self" title="MATRIXC1:def.7">Def7</a>, <a class="txt" href="matrixc1.html#E2:53_3"><i><font color="Green" title="E47">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">F</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<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="c3">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#T43" target="_self" title="MATRIXC1:th.43">Th43</a>, <a class="txt" href="matrixc1.html#E2:53_3"><i><font color="Green" title="E47">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<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="c3">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#T44" target="_self" title="MATRIXC1:th.44">Th44</a>, <a class="txt" href="matrixc1.html#E1:53"><i><font color="Green" title="E40">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<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="c3">i</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></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="matrixc1.html#T24" target="_self" title="MATRIXC1:th.24">Th24</a>, <a class="txt" href="matrixc1.html#E5:53_3"><i><font color="Green" title="E50">A11</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="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self" title="MATRIXC1:def.7">Def7</a>, <a class="txt" href="matrixc1.html#E3:53_3"><i><font color="Green" title="E48">A9</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:53"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">F</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
 <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#E6:53"><i><font color="Green" title="E45">A6</font></i></a>, <a class="txt" href="matrixc1.html#E2:53"><i><font color="Green" title="E41">A2</font></i></a></i>;<br/>


</div>
