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

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">z</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font> <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="c2">y</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> <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="c3">z</font> implies  <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font> <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="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> <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="c3">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">x2</font> = <font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c5">y2</font> = <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c6">z2</font> = <font color="Maroon" title="c3">z</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E33">A2</font></span></a>, <a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></span>;<br/>
<span class="lab"><font color="Green" title="E34">A3</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <font color="Maroon" title="c5">y2</font></span>)</span>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E35">A4</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<span class="lab"><font color="Green" title="E36">A5</font></span>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c5">y2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E7:45"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_4"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c2">y</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">d</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">e</font> = <font color="Maroon" title="c3">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font>;<br/>
<a NAME="E2:45_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <font color="Maroon" title="c5">y2</font></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">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<span class="kw">then </span> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x2</font> <a href="complsp2.html#K3" title="COMPLSP2:func.3">+</a> <font color="Maroon" title="c5">y2</font></span>)</span> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K8" title="MATRIXC1:func.8">mlt</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">z2</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>
;<br/>
<a NAME="E4:45_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A7</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="complsp2.html#T1" title="COMPLSP2:th.1">COMPLSP2:1</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="matrixc1.html#T19" target="_self" title="MATRIXC1:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="matrixc1.html#T19" target="_self" title="MATRIXC1:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="matrixc1.html#T19" target="_self" title="MATRIXC1:th.19">Th19</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c7">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="complsp2.html#T1" title="COMPLSP2:th.1">COMPLSP2:1</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:45"/>
 <a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">z</font></span>)</span> <a href="seq_4.html#K9" title="SEQ_4:func.9">+</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
