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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K2">COMPLEX</a> ;<br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green">E53</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:67"/><i><font color="Green">E54</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<i><font color="Green">E55</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
;<br/>
<i><font color="Green">E56</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>
;<br/>
<i><font color="Green">E57</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>
;<br/>
<a NAME="E6:67"/><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<i><font color="Green">E59</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </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">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E3:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E4:67"><i><font color="Green">E56</font></i></a></i>
;<br/>
<a NAME="E8:67"/><i><font color="Green">E60</font></i>: 
 <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E9:67"/><i><font color="Green">E61</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E10:67"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E9:67"><i><font color="Green">E61</font></i></a>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E11:67"/><i><font color="Green">E63</font></i>: 
 <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">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E3:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E4:67"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E12:67"/><i><font color="Green">E64</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrixc1.html#E8:67"><i><font color="Green">E60</font></i></a></i>;<br/>
<i><font color="Green">E65</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrixc1.html#E8:67"><i><font color="Green">E60</font></i></a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>
;<br/>
<i><font color="Green">E66</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> = 
 <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">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E3:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E4:67"><i><font color="Green">E56</font></i></a></i>
;<br/>
<a NAME="E15:67"/><i><font color="Green">E67</font></i>: 
 <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">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E3:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E4:67"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E16:67"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><a NAME="E1:67_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span>
 ;<br/>

<a NAME="E2:67_7"/><b>then </b><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E6:67"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrixc1.html#E12:67"><i><font color="Green">E64</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E3:67_7"/><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E2:67_7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E4:67_7"/><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">c<sub>3</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E2:67_7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E5:67_7"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:67_7"><i><font color="Green">E70</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E6:67_7"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <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">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E2:67_7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrixc1.html#E11:67"><i><font color="Green">E63</font></i></a>, <a class="txt" href="matrixc1.html#E15:67"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E7:67_7"/><i><font color="Green">E73</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E6:67_7"><i><font color="Green">E72</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:67_7"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E2:67_7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>;<br/>
<a NAME="E9:67_7"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E10:67"><i><font color="Green">E62</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E2:67_7"><i><font color="Green">E68</font></i></a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a>, <a class="txt" href="matrixc1.html#E4:67"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E10:67_7"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E9:67_7"><i><font color="Green">E75</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E11:67_7"/><i><font color="Green">E77</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></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"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E2:67_7"><i><font color="Green">E68</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self">Th1</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E7:67_7"><i><font color="Green">E73</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E10:67_7"><i><font color="Green">E76</font></i></a>, <a class="txt" href="matrixc1.html#E3:67"><i><font color="Green">E55</font></i></a>, <a class="txt" href="matrixc1.html#E4:67"><i><font color="Green">E56</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> 
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, <a class="txt" href="matrixc1.html#E8:67_7"><i><font color="Green">E74</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </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"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T121">COMPLEX1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </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"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a>, <a class="txt" href="matrixc1.html#E3:67_7"><i><font color="Green">E69</font></i></a>, <a class="txt" href="matrixc1.html#E2:67"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span></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"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T121">COMPLEX1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></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"><font color="Maroon">c<sub>1</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E10:67_7"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="complsp2.html#T22">COMPLSP2:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="txt" href="matrixc1.html#E5:67_7"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="binop_2.html#K5">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E5:67_7"><i><font color="Green">E71</font></i></a>, <a class="txt" href="matrixc1.html#E1:67"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="txt" href="matrixc1.html#E11:67_7"><i><font color="Green">E77</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:67_7"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E17:67"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K4">@</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="matrixc1.html#K1" target="_self">*'</a> 
 <b>by </b><i><a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a>, <a class="txt" href="matrixc1.html#E5:67"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrixc1.html#E7:67"><i><font color="Green">E59</font></i></a>, <a class="txt" href="matrixc1.html#E13:67"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrixc1.html#E14:67"><i><font color="Green">E66</font></i></a></i>;<br/>


</div>
