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

<b>let </b><font color="Maroon">x</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> , <font color="Maroon">y</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">M</font> be   <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>



<b>set </b><font color="Maroon">Z</font> =  <a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font>;<br/>
<b>assume </b><a NAME="E1:56"/><i><font color="Green">E43</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:56"/><i><font color="Green">E48</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E3:56"/><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:56"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrixr1.html#T61">MATRIXR1:61</a></i>;<br/>
<b>then </b><i><font color="Green">E50</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E5:56"><i><font color="Green">E50</font></i></a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrprob.html#T1" target="_self">Th1</a></i>
;<br/>
<a NAME="E6:56"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrprob.html#NK4" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:56_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:56_2"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" target="_self">LineSum</a> <span class="p3">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:56_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:56_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:56_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E5:56"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrprob.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:56_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:56_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrixr1.html#T61">MATRIXR1:61</a>, </i>;<br/>
<a NAME="E7:56_2"/><i><font color="Green">E62</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E8:56_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E77</font></i>: <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T86">RVSUM_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="matrprob.html#NK7" target="_self">"*"</a> <font color="Maroon">y</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a>, , </i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117">RVSUM_1:117</a></i>
;<br/>
<a NAME="E10:56_2"/>
 <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="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E3:56"><i><font color="Green">E49</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E11:56_2"/><b>then </b><i><font color="Green">E117</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E3:56_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="euclid_2.html#T8">EUCLID_2:8</a></i>;<br/>
<a NAME="E12:56_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_2_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:56_2_2"/><i><font color="Green">E118</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> )
 ;<br/>

<a NAME="E2:56_2_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p3">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="txt" href="matrprob.html#E6:56_2"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E3:56_2_2"/><b>then </b><i><font color="Green">E119</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E4:56_2_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E3:56"><i><font color="Green">E49</font></i></a>, <a class="ref" href="matrprob.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E5:56_2_2"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:56_2_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E6:56_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="matrprob.html#E3:56"><i><font color="Green">E49</font></i></a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E7:56_2_2"/><b>then </b><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="matrixc1.html#T1">MATRIXC1:1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</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> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E8:56_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="rvsum_1.html#T86">RVSUM_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</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="matrprob.html#E7:56_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="matrprob.html#E3:56"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green">E48</font></i></a>, <a class="ref" href="matrprob.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font>
<b>by </b><i><a class="txt" href="matrprob.html#E7:56_2"><i><font color="Green">E62</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="E13:56_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font></span>)</span> <a href="rvsum_1.html#K9">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>,<font color="Maroon">i</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E5:56_2"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>
<b>hence </b><a NAME="E14:56_2"/>
<span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" target="_self">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="matrprob.html#E4:56_2"><i><font color="Green">E59</font></i></a>, , <a class="txt" href="matrprob.html#E5:56"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrprob.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:56"/>
<span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrixr1.html#K11">*</a> <font color="Maroon">y</font></span>)</span></span><a href="euclid_2.html#K1">)|</a></span> <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K5" target="_self">SumAll</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">x</font>,<font color="Maroon">M</font>,<font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a>, </i>;<br/>


</div>
