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<b>let </b><font color="Maroon">a</font> be  <a href="xcmplx_0.html#V1">complex</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="xcmplx_0.html#D2">XCMPLX_0:def 2</a></i>;<br/>
<a NAME="E2:6"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:6"/><i><font color="Green">E36</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:6"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></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">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E5:6"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:6"/><b>then </b><i><font color="Green">E37</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:6"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E8:6"/><i><font color="Green">E40</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:6"/><i><font color="Green">E41</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E10:6"/>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></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">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:6"/><b>then </b>
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 
<b>by </b><i/>;<br/>
<a NAME="E12:6"/><b>then </b><i><font color="Green">E42</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></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"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E13:6"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 ;<br/><a NAME="E2:6_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> )
 <b>by </b><i>, , </i>;<br/><a NAME="E3:6_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:6_1"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">M</font>
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:6_1"/><i><font color="Green">E56</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</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">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 <b>by </b><i>, , , , , </i>;<br/><a NAME="E6:6_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="complex1.html#K15">*'</a> 
 <b>by </b><i>, </i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">aa</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T121">COMPLEX1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E15:6"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="matrix_5.html#K7">*</a> <font color="Maroon">M</font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="matrix_5.html#K7">*</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span>
 <b>by </b><i><a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a>, , </i>;<br/>


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