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

<b>assume </b><a NAME="E1:46"/>
(  <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> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">z</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font>, <font color="Maroon">y2</font> = <font color="Maroon">y</font>, <font color="Maroon">z2</font> = <font color="Maroon">z</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/>
<i><font color="Green">E35</font></i>:  <a href="finseq_1.html#K4">dom</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="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <span class="p3">(<span class="default"><font color="Maroon">x2</font> <a href="complsp2.html#K3">+</a> <font color="Maroon">y2</font></span>)</span>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span> <a href="complsp2.html#K3">+</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span> <a href="complsp2.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E36</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span> <a href="complsp2.html#K3">+</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span> <a href="complsp2.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<i><font color="Green">E37</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span> <a href="complsp2.html#K3">+</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span> <a href="complsp2.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E6:46"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span>)</span> holds <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="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></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"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></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="46_4"><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/>

<b>assume </b><a NAME="E1:46_4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></span>)</span>
 ;<br/>

<a NAME="E2:46_4"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="complsp2.html#K3">+</a> <font color="Maroon">y2</font></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="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<b>then </b><i><font color="Green">E40</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x2</font> <a href="complsp2.html#K3">+</a> <font color="Maroon">y2</font></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K9" target="_self">mlt</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <font color="Maroon">y</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">d</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>;<br/>
<b>set </b><font color="Maroon">e</font> = <font color="Maroon">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>;<br/>
<a NAME="E4:46_4"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K3">+</a> <font color="Maroon">y</font></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"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</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="txt" href="matrixc1.html#E3:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrixc1.html#E4:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrixc1.html#E1:46_4"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="complsp2.html#T1">COMPLSP2:1</a></i>;<br/>
<b>thus </b><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="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font></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"><font color="Maroon">x</font> <a href="complsp1.html#K3">+</a> <font color="Maroon">y</font></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">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:46_4"><i><font color="Green">E38</font></i></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">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><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> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E3:46_4"><i><font color="Green">E40</font></i></a></i>
<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">z</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</a> <span class="p1">(<span class="default"><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> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E3:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrixc1.html#E4:46"><i><font color="Green">E36</font></i></a>, <a class="txt" href="matrixc1.html#E1:46_4"><i><font color="Green">E38</font></i></a>, </i>
<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">z</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K3">+</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">y</font>,<font color="Maroon">z</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E3:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrixc1.html#E5:46"><i><font color="Green">E37</font></i></a>, <a class="txt" href="matrixc1.html#E1:46_4"><i><font color="Green">E38</font></i></a>, </i>
<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">z</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E3:46"><i><font color="Green">E35</font></i></a>, <a class="txt" href="matrixc1.html#E1:46_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="complsp2.html#T1">COMPLSP2:1</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:46"/>
 <a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="complsp1.html#K3">+</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="complsp1.html#K3">+</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E3:46"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>;<br/>


</div>
