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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 ;<br/>

<i><font color="Green">E43</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self">Def7</a></i>
;<br/>
<a NAME="E3:54"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:54"/><i><font color="Green">E45</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E5:54"/><i><font color="Green">E46</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<i><font color="Green">E47</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self">Def7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:54_3"/><i><font color="Green">E48</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> )
 ;<br/><a NAME="E2:54_3"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:54_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E2:54"><i><font color="Green">E43</font></i></a></i>;<br/><a NAME="E3:54_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:54_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E2:54"><i><font color="Green">E43</font></i></a>, <a class="txt" href="matrixc1.html#E3:54"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E4:54_3"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E5:54"><i><font color="Green">E46</font></i></a>, <a class="txt" href="matrixc1.html#E1:54"><i><font color="Green">E42</font></i></a>, <a class="txt" href="matrixc1.html#E4:54"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E5:54_3"/><i><font color="Green">E52</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, <a class="txt" href="matrixc1.html#E4:54_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="matrixc1.html#E4:54"><i><font color="Green">E45</font></i></a>, <a class="txt" href="matrixc1.html#E1:54"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E6:54_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:54_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self">Def7</a>, <a class="txt" href="matrixc1.html#E2:54_3"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#T43" target="_self">Th43</a>, <a class="txt" href="matrixc1.html#E2:54_3"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#T44" target="_self">Th44</a>, <a class="txt" href="matrixc1.html#E1:54"><i><font color="Green">E42</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#T24" target="_self">Th24</a>, <a class="txt" href="matrixc1.html#E5:54_3"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D7" target="_self">Def7</a>, <a class="txt" href="matrixc1.html#E3:54_3"><i><font color="Green">E50</font></i></a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:54"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E6:54"><i><font color="Green">E47</font></i></a>, <a class="txt" href="matrixc1.html#E2:54"><i><font color="Green">E43</font></i></a></i>;<br/>


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