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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</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> ;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:71"/><i><font color="Green" title="E53">A1</font></i>: 
(  <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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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">M</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:71"/><b>then </b><i><font color="Green" title="E54">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/>
<i><font color="Green" title="E55">A3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a>, <a class="txt" href="matrixc1.html#E1:71"><i><font color="Green" title="E53">A1</font></i></a></i>
;<br/>
<a NAME="E4:71"/><i><font color="Green" title="E56">A4</font></i>: 
0 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:71"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:71"/><i><font color="Green" title="E57">A5</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="c2">y</font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:71"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<a NAME="E6:71"/><b>then </b><i><font color="Green" title="E58">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </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="c2">y</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E7:71"/><i><font color="Green" title="E59">A7</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</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>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:71"><i><font color="Green" title="E55">A3</font></i></a>, <a class="ref" href="matrixc1.html#D7" target="_self" title="MATRIXC1:def.7">Def7</a></i>;<br/>
<a NAME="E8:71"/><b>then </b><i><font color="Green" title="E60">A8</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </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>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>;<br/>
<i><font color="Green" title="E61">A9</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></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#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<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#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
;<br/>
<i><font color="Green" title="E62">A10</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:71"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="matrixc1.html#E5:71"><i><font color="Green" title="E57">A5</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>
;<br/>
<a NAME="E11:71"/><b>then </b><i><font color="Green" title="E63">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></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> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E8:71"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E12:71"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></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"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </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="71_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71_4"/><i><font color="Green" title="E64">A12</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:71_4"/><b>then </b>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:71_4"/><b>then </b><i><font color="Green" title="E65">A13</font></i>: 
<font color="Maroon" title="c4">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#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:71_4"/>
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E11:71"><i><font color="Green" title="E63">A11</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:71_4"><i><font color="Green" title="E64">A12</font></i></a></i>;<br/>
<a NAME="E5:71_4"/><b>then </b>
<font color="Maroon" title="c4">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> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green" title="E66">A14</font></i>: <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> = 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T19" target="_self" title="MATRIXC1:th.19">Th19</a></i>
<br/>.= 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</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="c4">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:71_4"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="matrixc1.html#E7:71"><i><font color="Green" title="E59">A7</font></i></a>, <a class="txt" href="matrixc1.html#E10:71"><i><font color="Green" title="E62">A10</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
;<br/>
<a NAME="E7:71_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a>, <a class="txt" href="matrixc1.html#E2:71"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="matrixc1.html#E5:71"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/>
<a NAME="E8:71_4"/><b>then </b><i><font color="Green" title="E67">A15</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>;<br/>
<a NAME="E9:71_4"/><b>then </b><i><font color="Green" title="E68">A16</font></i>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:71_4"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="matrixc1.html#E2:71"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="matrixc1.html#E3:71"><i><font color="Green" title="E55">A3</font></i></a></i>;<br/>
<a NAME="E10:71_4"/><i><font color="Green" title="E69">A17</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</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="c2">y</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E6:71"><i><font color="Green" title="E58">A6</font></i></a></i>;<br/>
<a NAME="E11:71_4"/><b>then </b><i><font color="Green" title="E70">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:71"><i><font color="Green" title="E56">A4</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/>
<i><font color="Green" title="E71">A19</font></i>: <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> = 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:71_4"><i><font color="Green" title="E66">A14</font></i></a>, <a class="txt" href="matrixc1.html#E9:71_4"><i><font color="Green" title="E68">A16</font></i></a>, <a class="ref" href="matrixc1.html#D7" target="_self" title="MATRIXC1:def.7">Def7</a></i>
<br/>.= 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T24" target="_self" title="MATRIXC1:th.24">Th24</a>, <a class="txt" href="matrixc1.html#E11:71_4"><i><font color="Green" title="E70">A18</font></i></a></i>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <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="c4">i</font></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T29" target="_self" title="MATRIXC1:th.29">Th29</a></i>
;<br/>
<i><font color="Green" title="E72">A20</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a>, <a class="txt" href="matrixc1.html#E2:71"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="matrixc1.html#E5:71"><i><font color="Green" title="E57">A5</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E14:71_4"/><i><font color="Green" title="E73">A21</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</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="c2">y</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E6:71"><i><font color="Green" title="E58">A6</font></i></a></i>;<br/>
<i><font color="Green" title="E74">A22</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <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="c4">i</font></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T3" title="COMPLSP2:th.3">COMPLSP2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E13:71_4"><i><font color="Green" title="E72">A20</font></i></a>, <a class="txt" href="matrixc1.html#E14:71_4"><i><font color="Green" title="E73">A21</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>
;<br/>
<a NAME="E16:71_4"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> holds <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="c4">i</font></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:71_4_5"/><i><font color="Green" title="E75">A23</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> )
 ;<br/>

<a NAME="E2:71_4_5"/><b>then </b>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:71_4_5"/><b>then </b><i><font color="Green" title="E76">A24</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<i><font color="Green" title="E77">A25</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:71"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
;<br/>
<a NAME="E5:71_4_5"/>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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="c2">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:71_4_5"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="matrixc1.html#E13:71_4"><i><font color="Green" title="E72">A20</font></i></a></i>;<br/>
<a NAME="E6:71_4_5"/><b>then </b>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E7:71_4_5"/><b>then </b>
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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#K5" title="MATRIXC1:func.5">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:71_4_5"><i><font color="Green" title="E77">A25</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E8:71_4_5"/><b>then </b><i><font color="Green" title="E78">A26</font></i>: 
<font color="Maroon" title="c5">j</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"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:71_4_5"/><i><font color="Green" title="E79">A27</font></i>: 
<font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:71_4_5"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="matrixc1.html#E13:71_4"><i><font color="Green" title="E72">A20</font></i></a>, <a class="txt" href="matrixc1.html#E5:71"><i><font color="Green" title="E57">A5</font></i></a></i>;<br/>
<a NAME="E10:71_4_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:71_4_5"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="matrixc1.html#E13:71_4"><i><font color="Green" title="E72">A20</font></i></a>, <a class="txt" href="matrixc1.html#E1:71"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<a NAME="E11:71_4_5"/><b>then </b><i><font color="Green" title="E80">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:71_4"><i><font color="Green" title="E64">A12</font></i></a>, <a class="txt" href="matrixc1.html#E8:71_4"><i><font color="Green" title="E67">A15</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<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="c4">i</font></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font> = 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T16" title="COMPLSP2:th.16">COMPLSP2:16</a></i>
<br/>.= 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T28" target="_self" title="MATRIXC1:th.28">Th28</a>, <a class="txt" href="matrixc1.html#E10:71_4"><i><font color="Green" title="E69">A17</font></i></a></i>
<br/>.= 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T43" target="_self" title="MATRIXC1:th.43">Th43</a>, <a class="txt" href="matrixc1.html#E2:71"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="matrixc1.html#E3:71"><i><font color="Green" title="E55">A3</font></i></a>, <a class="txt" href="matrixc1.html#E9:71_4"><i><font color="Green" title="E68">A16</font></i></a></i>
<br/>.= 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T19" target="_self" title="MATRIXC1:th.19">Th19</a>, <a class="txt" href="matrixc1.html#E8:71_4_5"><i><font color="Green" title="E78">A26</font></i></a></i>
<br/>.= 
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a>, <a class="txt" href="matrixc1.html#E1:71_4_5"><i><font color="Green" title="E75">A23</font></i></a>, <a class="txt" href="matrixc1.html#E13:71_4"><i><font color="Green" title="E72">A20</font></i></a></i>
<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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</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="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</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="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E9:71_4_5"><i><font color="Green" title="E79">A27</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a>, <a class="txt" href="matrixc1.html#E2:71"><i><font color="Green" title="E54">A2</font></i></a>, <a class="txt" href="matrixc1.html#E5:71"><i><font color="Green" title="E57">A5</font></i></a>, <a class="txt" href="matrixc1.html#E11:71_4_5"><i><font color="Green" title="E80">A28</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:71_4_5"/>
<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="c4">i</font></span>)</span> <a href="complsp1.html#K6" title="COMPLSP1:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E3:71_4_5"><i><font color="Green" title="E76">A24</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E17:71_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="matrixc1.html#K5" title="MATRIXC1:func.5">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c4">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E12:71_4"><i><font color="Green" title="E71">A19</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E15:71_4"><i><font color="Green" title="E74">A22</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:71_4"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</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>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font>
 <b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a>, <a class="txt" href="matrixc1.html#E9:71"><i><font color="Green" title="E61">A9</font></i></a>, <a class="txt" href="matrixc1.html#E3:71_4"><i><font color="Green" title="E65">A13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:71"/><b>then </b>
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</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>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E11:71"><i><font color="Green" title="E63">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E14:71"/>
<span class="p1"><a href="complsp2.html#K8" title="COMPLSP2:func.8">|(</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K2" title="MATRIXC1:func.2">@"</a> </span>)</span> <a href="matrixc1.html#K6" title="MATRIXC1:func.6">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="complsp2.html#K8" title="COMPLSP2:func.8">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixc1.html#K12" title="MATRIXC1:func.12">SumAll</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrixc1.html#T40" target="_self" title="MATRIXC1:th.40">Th40</a>, <a class="txt" href="matrixc1.html#E1:71"><i><font color="Green" title="E53">A1</font></i></a>, <a class="txt" href="matrixc1.html#E7:71"><i><font color="Green" title="E59">A7</font></i></a></i>;<br/>


</div>
