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<b>let </b><font color="Maroon" title="c1">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/>

<a NAME="E1:3"/><i><font color="Green" title="E4">A1</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="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E2:3"/><i><font color="Green" title="E5">A2</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">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="c1">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="E3:3"/><i><font color="Green" title="E6">A3</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E4:3"/><i><font color="Green" title="E7">A4</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="c1">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="c1">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="3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 ;<br/><a NAME="E2:3_1"/><b>then </b><i><font color="Green" title="E8">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">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="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> &amp; 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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="matrixc1.html#E3:3"><i><font color="Green" title="E6">A3</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/><a NAME="E3:3_1"/><b>then </b><i><font color="Green" title="E9">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/><a NAME="E4:3_1"/><i><font color="Green" title="E10">A7</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">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="c1">M</font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="matrixc1.html#E4:3"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="matrixc1.html#E2:3_1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/><a NAME="E5:3_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c2">i</font>,<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">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#E4:3_1"><i><font color="Green" title="E10">A7</font></i></a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <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="c2">i</font>,<font color="Maroon" title="c3">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">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="txt" href="matrixc1.html#E3:3_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
<br/>.= 
<font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>

;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a>, <a class="txt" href="matrixc1.html#E1:3"><i><font color="Green" title="E4">A1</font></i></a>, <a class="txt" href="matrixc1.html#E2:3"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="matrixc1.html#E3:3"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="matrixc1.html#E4:3"><i><font color="Green" title="E7">A4</font></i></a></i>;<br/>


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