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

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

<a NAME="E1:3"/><i><font color="Green">E4</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>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </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="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:3"/><i><font color="Green">E5</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="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>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:3"/><i><font color="Green">E6</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green">E7</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</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>1</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="3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:3_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 ;<br/><a NAME="E2:3_1"/><b>then </b><i><font color="Green">E8</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</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>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> &amp; 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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="matrixc1.html#E3:3"><i><font color="Green">E6</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E3:3_1"/><b>then </b><i><font color="Green">E9</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E4:3_1"/><i><font color="Green">E10</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</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>1</sub></font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E2:3"><i><font color="Green">E5</font></i></a>, <a class="txt" href="matrixc1.html#E4:3"><i><font color="Green">E7</font></i></a>, <a class="txt" href="matrixc1.html#E2:3_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:3_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>2</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#E4:3_1"><i><font color="Green">E10</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>,<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>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>2</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="txt" href="matrixc1.html#E3:3_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>

;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a>, <a class="txt" href="matrixc1.html#E1:3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="matrixc1.html#E2:3"><i><font color="Green">E5</font></i></a>, <a class="txt" href="matrixc1.html#E3:3"><i><font color="Green">E6</font></i></a>, <a class="txt" href="matrixc1.html#E4:3"><i><font color="Green">E7</font></i></a></i>;<br/>


</div>
