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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</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="E1:52"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>1</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>
 ;<br/>

<a NAME="E2:52"/><i><font color="Green">E41</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></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="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<i><font color="Green">E42</font></i>:  <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#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>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"><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>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <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/>
<a NAME="E4:52"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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>1</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_2"><b>proof </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="E1:52_2"/><i><font color="Green">E43</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"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:52_2"/><i><font color="Green">E44</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="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:52_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E3:52_2"/><i><font color="Green">E45</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"><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>1</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:52_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E3:52"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:52_2"/><i><font color="Green">E46</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"><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>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:52_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E5:52_2"/><i><font color="Green">E47</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</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="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:52"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E6:52_2"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E5:52_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="matrixc1.html#E2:52_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:52_2"/><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="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:52_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E2:52"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E8:52_2"/><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="matrix_1.html#K1">width</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#E7:52_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<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 class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E7:52_2"><i><font color="Green">E49</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="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E6:52_2"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><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>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="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E8:52_2"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><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>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>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, <a class="txt" href="matrixc1.html#E4:52_2"><i><font color="Green">E46</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E10:52_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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>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>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:52"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</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>1</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E2:52"><i><font color="Green">E41</font></i></a>, <a class="txt" href="matrixc1.html#E3:52"><i><font color="Green">E42</font></i></a></i>;<br/>


</div>
