<?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="E1:72"/><i><font color="Green">E57</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="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>2</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>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="E2:72"/><i><font color="Green">E58</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="matrix_1.html#K4">@</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="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="txt" href="matrixc1.html#E1:72"><i><font color="Green">E57</font></i></a></i>;<br/>
<i><font color="Green">E59</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>1</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#E1:72"><i><font color="Green">E57</font></i></a></i>
;<br/>
<a NAME="E4:72"/><i><font color="Green">E60</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">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:72"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:72"/><i><font color="Green">E61</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="matrix_1.html#K4">@</a> </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="txt" href="matrixc1.html#E1:72"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<a NAME="E6:72"/><i><font color="Green">E62</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">c<sub>3</sub></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">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self">Def1</a>, <a class="txt" href="matrixc1.html#E5:72"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E7:72"/><i><font color="Green">E63</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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <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>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E3:72"><i><font color="Green">E59</font></i></a>, <a class="ref" href="matrixc1.html#D7" target="_self">Def7</a></i>;<br/>
<a NAME="E8:72"/><i><font color="Green">E64</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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E7:72"><i><font color="Green">E63</font></i></a>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>;<br/>
<i><font color="Green">E65</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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a></i>
;<br/>
<i><font color="Green">E66</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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</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="txt" href="matrixc1.html#E2:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrixc1.html#E5:72"><i><font color="Green">E61</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self">Def13</a></i>
;<br/>
<a NAME="E11:72"/><b>then </b><i><font color="Green">E67</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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E8:72"><i><font color="Green">E64</font></i></a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E12:72"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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#K11" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:72_4"/><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> <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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:72_4"/>
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E3:72_4"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>4</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E11:72"><i><font color="Green">E67</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E5:72_4"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></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">E72</font></i>: <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E5:72_4"><i><font color="Green">E71</font></i></a>, <a class="ref" href="matrixc1.html#T19" target="_self">Th19</a></i>
<br/>.= 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span> <a href="matrixc1.html#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <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>
<b>by </b><i><a class="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="matrixc1.html#E7:72"><i><font color="Green">E63</font></i></a>, <a class="txt" href="matrixc1.html#E10:72"><i><font color="Green">E66</font></i></a>, <a class="ref" href="complsp2.html#D1">COMPLSP2:def 1</a></i>
;<br/>
<a NAME="E7:72_4"/><i><font color="Green">E73</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="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>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</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#E2:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrixc1.html#E5:72"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E8:72_4"/><i><font color="Green">E74</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="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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a>, <a class="txt" href="matrixc1.html#E7:72_4"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E9:72_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></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="txt" href="matrixc1.html#E1:72_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="matrixc1.html#E2:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrixc1.html#E3:72"><i><font color="Green">E59</font></i></a>, <a class="txt" href="matrixc1.html#E8:72_4"><i><font color="Green">E74</font></i></a></i>;<br/>
<a NAME="E10:72_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></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><a class="txt" href="matrixc1.html#E9:72_4"><i><font color="Green">E75</font></i></a></i>;<br/>
<a NAME="E11:72_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E10:72_4"><i><font color="Green">E76</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E12:72_4"/><i><font color="Green">E78</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="matrix_1.html#K4">@</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E11:72_4"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E79</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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></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="matrixc1.html#K2" target="_self">@"</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">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:72"><i><font color="Green">E62</font></i></a></i>
;<br/>
<a NAME="E14:72_4"/><i><font color="Green">E80</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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:72"><i><font color="Green">E60</font></i></a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a>, <a class="txt" href="matrixc1.html#E13:72_4"><i><font color="Green">E79</font></i></a></i>;<br/>
<i><font color="Green">E81</font></i>: <span class="p1">(<span class="default"><a href="matrixc1.html#K5" target="_self">mlt</a> <font color="Maroon">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="complex1.html#K15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:72_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="matrixc1.html#E9:72_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="matrixc1.html#D7" target="_self">Def7</a></i>
<br/>.= 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T24" target="_self">Th24</a>, <a class="txt" href="matrixc1.html#E14:72_4"><i><font color="Green">E80</font></i></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">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrixc1.html#T29" target="_self">Th29</a></i>
;<br/>
<i><font color="Green">E82</font></i>:  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</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#E2:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrixc1.html#E5:72"><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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>
;<br/>
<i><font color="Green">E83</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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></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="matrixc1.html#K2" target="_self">@"</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">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="matrixc1.html#E6:72"><i><font color="Green">E62</font></i></a></i>
;<br/>
<i><font color="Green">E84</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">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E16:72_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="matrixc1.html#E17:72_4"><i><font color="Green">E83</font></i></a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a></i>
;<br/>
<a NAME="E19:72_4"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</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="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> holds <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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_4_7"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:72_4_7"/><i><font color="Green">E86</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> <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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:72_4_7"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:72_4_7"><i><font color="Green">E86</font></i></a></i>;<br/>
<a NAME="E3:72_4_7"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrixc1.html#E2:72_4_7"><i><font color="Green">E87</font></i></a></i>;<br/>
<i><font color="Green">E89</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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></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="matrix_1.html#D8">MATRIX_1:def 8</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#E1:72"><i><font color="Green">E57</font></i></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">c<sub>2</sub></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_7"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrixc1.html#E1:72_4_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="matrixc1.html#E16:72_4"><i><font color="Green">E82</font></i></a></i>;<br/>
<a NAME="E6:72_4_7"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></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_7"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:72_4_7"><i><font color="Green">E89</font></i></a>, <a class="ref" href="matrixc1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E8:72_4_7"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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_7"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">c<sub>5</sub></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">c<sub>3</sub></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="txt" href="matrixc1.html#E1:72_4_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="matrixc1.html#E16:72_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="matrixc1.html#E5:72"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E10:72_4_7"/><i><font color="Green">E92</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="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="txt" href="matrixc1.html#E1:72_4_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="matrixc1.html#E16:72_4"><i><font color="Green">E82</font></i></a>, <a class="txt" href="matrixc1.html#E1:72"><i><font color="Green">E57</font></i></a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<a NAME="E11:72_4_7"/><i><font color="Green">E93</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="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E10:72_4_7"><i><font color="Green">E92</font></i></a>, <a class="txt" href="matrixc1.html#E12:72_4"><i><font color="Green">E78</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"><font color="Maroon">c<sub>1</sub></font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></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#T16">COMPLSP2:16</a></i>
<br/>.= 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></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#T28" target="_self">Th28</a>, <a class="txt" href="matrixc1.html#E13:72_4"><i><font color="Green">E79</font></i></a></i>
<br/>.= 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="complsp2.html#K1">*'</a> </span>)</span></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#T43" target="_self">Th43</a>, <a class="txt" href="matrixc1.html#E2:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrixc1.html#E3:72"><i><font color="Green">E59</font></i></a>, <a class="txt" href="matrixc1.html#E9:72_4"><i><font color="Green">E75</font></i></a></i>
<br/>.= 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</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>
<b>by </b><i><a class="ref" href="matrixc1.html#T19" target="_self">Th19</a>, <a class="txt" href="matrixc1.html#E8:72_4_7"><i><font color="Green">E90</font></i></a></i>
<br/>.= 
<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>4</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <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"><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>5</sub></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="txt" href="matrixc1.html#E1:72_4_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="matrixc1.html#E16:72_4"><i><font color="Green">E82</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>4</sub></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">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <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"><font color="Maroon">c<sub>2</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>

<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>4</sub></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">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"><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>5</sub></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>, <a class="txt" href="matrixc1.html#E9:72_4_7"><i><font color="Green">E91</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K14" target="_self">QuadraticForm</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</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>
<b>by </b><i><a class="ref" href="matrixc1.html#D13" target="_self">Def13</a>, <a class="txt" href="matrixc1.html#E2:72"><i><font color="Green">E58</font></i></a>, <a class="txt" href="matrixc1.html#E5:72"><i><font color="Green">E61</font></i></a>, <a class="txt" href="matrixc1.html#E11:72_4_7"><i><font color="Green">E93</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E13:72_4_7"/>
<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>4</sub></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">c<sub>3</sub></font> <a href="matrixc1.html#K2" target="_self">@"</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="matrixc1.html#E3:72_4_7"><i><font color="Green">E88</font></i></a>, <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">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E15:72_4"><i><font color="Green">E81</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, <a class="txt" href="matrixc1.html#E18:72_4"><i><font color="Green">E84</font></i></a>, <a class="txt" href="matrixc1.html#E19:72_4"><i><font color="Green">E85</font></i></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="complsp2.html#K1">*'</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self">Def10</a>, <a class="txt" href="matrixc1.html#E9:72"><i><font color="Green">E65</font></i></a>, <a class="txt" href="matrixc1.html#E3:72_4"><i><font color="Green">E70</font></i></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">c<sub>1</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></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="txt" href="matrixc1.html#E11:72"><i><font color="Green">E67</font></i></a>, <a class="txt" href="matrixc1.html#E12:72"><i><font color="Green">E68</font></i></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">c<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<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#K7" target="_self">*</a> <font color="Maroon">c<sub>2</sub></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">c<sub>1</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="matrixc1.html#T40" target="_self">Th40</a>, <a class="txt" href="matrixc1.html#E1:72"><i><font color="Green">E57</font></i></a>, <a class="txt" href="matrixc1.html#E7:72"><i><font color="Green">E63</font></i></a></i>;<br/>


</div>
