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<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><a NAME="E1:33"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:33"/><i><font color="Green">E36</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> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</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"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E3:33"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">y</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">y</font>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green">E38</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x</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">x</font>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green">E40</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> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:33"/><i><font color="Green">E41</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> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</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">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E7:33"/><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="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i>, , , , </i>;<br/>
<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a>  as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">y1'</font> = <font color="Maroon">y</font> as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1'</font> = <font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a>  as    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="finseq_2.html#K4">-tuples_on</a> <a href="numbers.html#K2">COMPLEX</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110">FINSEQ_2:110</a></i>;<br/>
<b>set </b><font color="Maroon">a'</font> =  <a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1"/>
( 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"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> )
 ;<br/><a NAME="E2:33_1"/><b>then </b><i><font color="Green">E43</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"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> )
 <b>by </b><i><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"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</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"><span class="p2">(<span class="default"><font color="Maroon">x'</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">y'</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</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"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, <a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T22">COMPLSP2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, <a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x1'</font>,<font color="Maroon">y1'</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>

;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:33"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="complsp2.html#K1">*'</a>  <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">x</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>, </i>;<br/>


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