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

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:72"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:72"/><i><font color="Green">E36</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="ref" href="matrixc1.html#T6" target="_self">Th6</a></i>;<br/>
<i><font color="Green">E37</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="ref" href="matrixc1.html#T6" target="_self">Th6</a></i>
;<br/>
<a NAME="E4:72"/><i><font color="Green">E38</font></i>: 
0 <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:72"/><i><font color="Green">E40</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green">E41</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:72"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:72"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<i><font color="Green">E53</font></i>:  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>
;<br/>
<i><font color="Green">E54</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, , </i>
;<br/>
<a NAME="E11:72"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:72"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:72_4"/><i><font color="Green">E61</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:72_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E3:72_4"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:72_4"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T10" target="_self">Th10</a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E5:72_4"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green">E64</font></i>: <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</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">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</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">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T13" target="_self">Th13</a>, <a class="txt" href="matrixc1.html#E5:72_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="matrixc1.html#T9" target="_self">Th9</a>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
;<br/>
<a NAME="E7:72_4"/><i><font color="Green">E65</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, <a class="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E8:72_4"/><i><font color="Green">E103</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, </i>;<br/>
<a NAME="E9:72_4"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T13" target="_self">Th13</a>, <a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, , </i>;<br/>
<a NAME="E10:72_4"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E11:72_4"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E12:72_4"/><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E13:72_4"/><i><font color="Green">E108</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E3:72_4"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E14:72_4"/><i><font color="Green">E109</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i>, , </i>;<br/>
<i><font color="Green">E110</font></i>: <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrixc1.html#D7" target="_self">Def7</a></i>
<br/>.= 
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i/>
;<br/>
<i><font color="Green">E111</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, <a class="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<a NAME="E17:72_4"/><i><font color="Green">E113</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E3:72_4"><i><font color="Green">E62</font></i></a></i>;<br/>
<i><font color="Green">E114</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T3">COMPLSP2:3</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , </i>
;<br/>
<a NAME="E19:72_4"/><i><font color="Green">E115</font></i>: 
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:72_4_5"/><i><font color="Green">E116</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> )
 ;<br/>

<a NAME="E2:72_4_5"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E3:72_4_5"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="ref" href="matrixc1.html#T19" target="_self">Th19</a></i>;<br/>
<i><font color="Green">E119</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
<b>by </b><i><a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
;<br/>
<a NAME="E5:72_4_5"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, </i>;<br/>
<a NAME="E6:72_4_5"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E7:72_4_5"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:72_4_5"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E9:72_4_5"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, , <a class="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E10:72_4_5"/><i><font color="Green">E122</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, , <a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<a NAME="E11:72_4_5"/><i><font color="Green">E156</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T21" target="_self">Th21</a>, , </i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T16">COMPLSP2:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</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">Line</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="matrixc1.html#T20" target="_self">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</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">Line</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">y</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="matrixc1.html#T7" target="_self">Th7</a>, <a class="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E61</font></i></a>, </i>
;<br/>
<b>hence </b><a NAME="E13:72_4_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font></span>)</span> <a href="complsp1.html#K6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p5">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E20:72_4"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E23"><i><font color="Green">Lemma84</font></i></a>, <a class="ref" href="matrixc1.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E21:72_4"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self">Th1</a>, <a class="txt" href="matrixc1.html#E7:72_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="matrixc1.html#T14" target="_self">Th14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:72"/>
 <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="ref" href="matrixc1.html#T10" target="_self">Th10</a>, <a class="ref" href="matrixc1.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E14:72"/>
<span class="p1"><a href="complsp2.html#K8">|(</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">y</font></span>)</span></span><a href="complsp2.html#K8">)|</a></span> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K13" target="_self">SumAll</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E4:72"><i><font color="Green">E38</font></i></a>, <a class="ref" href="matrixc1.html#T6" target="_self">Th6</a>, </i>;<br/>


</div>
