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

<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="E1:36"/><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="E2:36"/><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"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></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">x</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E3:36"/><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="E4:36"/><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="E5:36"/><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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span>,<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">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E6:36"/><i><font color="Green">E41</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>,<font color="Maroon">y</font></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> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span>,<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="matrixc1.html#D1" target="_self">Def1</a>, , , </i>;<br/>
<a NAME="E7:36"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">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>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> holds <br/><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>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span>,<span class="p2">(<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="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </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/>

<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>,<font color="Maroon">y</font></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">y</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <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>
<b>by </b><i><a class="ref" href="complsp2.html#T22">COMPLSP2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span>,<span class="p2">(<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>
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E2:36_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>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> implies <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>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span>,<span class="p2">(<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> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:36"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a>  <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp2.html#K1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">y</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/>


</div>
