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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c3">M</font> be   <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>set </b><font color="Maroon" title="c4">Z</font> =  <a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font>;<br/>
<b>assume </b><a NAME="E1:56"/><i><font color="Green" title="E48">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 )
 ;<br/>

<a NAME="E2:56"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#D4" target="_self" title="MATRPROB:def.4">Def4</a></i>;<br/>
<a NAME="E3:56"/><i><font color="Green" title="E50">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="matrprob.html#T20" target="_self" title="MATRPROB:th.20">Th20</a></i>;<br/>
<a NAME="E4:56"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="matrixr1.html#T61" title="MATRIXR1:th.61">MATRIXR1:61</a></i>;<br/>
<b>then </b><i><font color="Green" title="E51">A4</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">x</font>
<b>by </b><i><a class="ref" href="matrprob.html#T30" target="_self" title="MATRPROB:th.30">Th30</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="matrprob.html#D4" target="_self" title="MATRPROB:def.4">Def4</a></i>
;<br/>
<a NAME="E6:56"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:56_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> )
 ;<br/>

<a NAME="E2:56_2"/><b>then </b><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:56_2"/><b>then </b><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E5:56"><i><font color="Green" title="E51">A4</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E4:56_2"/><i><font color="Green" title="E54">A7</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E3:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="matrprob.html#D4" target="_self" title="MATRPROB:def.4">Def4</a>, <a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="matrprob.html#E2:56_2"><i><font color="Green" title="E52">A5</font></i></a></i>;<br/>
<a NAME="E5:56_2"/><b>then </b><i><font color="Green" title="E55">A8</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:56_2"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="matrixr1.html#T61" title="MATRIXR1:th.61">MATRIXR1:61</a>, <a class="txt" href="matrprob.html#E4:56_2"><i><font color="Green" title="E54">A7</font></i></a></i>;<br/>
<a NAME="E7:56_2"/><i><font color="Green" title="E57">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E8:56_2"/><b>then </b><i><font color="Green" title="E58">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T30" target="_self" title="MATRPROB:th.30">Th30</a></i>;<br/>
<i><font color="Green" title="E59">A12</font></i>: <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:56_2"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="rvsum_1.html#T86" title="RVSUM_1:th.86">RVSUM_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <font color="Maroon" title="c2">y</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="matrprob.html#E6:56_2"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="matrprob.html#T41" target="_self" title="MATRPROB:th.41">Th41</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T117" title="RVSUM_1:th.117">RVSUM_1:117</a></i>
;<br/>
<a NAME="E10:56_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E11:56_2"/><b>then </b><i><font color="Green" title="E60">A13</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E8:56_2"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="euclid_2.html#T8" title="EUCLID_2:th.8">EUCLID_2:8</a></i>;<br/>
<a NAME="E12:56_2"/>
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">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" title="c6">j</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:56_2_2"/><i><font color="Green" title="E61">A14</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> )
 ;<br/>

<a NAME="E2:56_2_2"/>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p3">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="txt" href="matrprob.html#E1:56_2_2"><i><font color="Green" title="E61">A14</font></i></a></i>;<br/>
<a NAME="E3:56_2_2"/><b>then </b><i><font color="Green" title="E62">A15</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E4:56_2_2"/><b>then </b>
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="matrprob.html#E7:56_2"><i><font color="Green" title="E57">A10</font></i></a>, <a class="ref" href="matrprob.html#T30" target="_self" title="MATRPROB:th.30">Th30</a></i>;<br/>
<a NAME="E5:56_2_2"/><b>then </b><i><font color="Green" title="E63">A16</font></i>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:56_2_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:56_2_2"><i><font color="Green" title="E61">A14</font></i></a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a></i>;<br/>
<a NAME="E7:56_2_2"/><b>then </b><i><font color="Green" title="E64">A17</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c3">M</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E5:56_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="matrixc1.html#T1" title="MATRIXC1:th.1">MATRIXC1:1</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T66" title="RVSUM_1:th.66">RVSUM_1:66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">j</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E5:56_2_2"><i><font color="Green" title="E63">A16</font></i></a>, <a class="ref" href="rvsum_1.html#T86" title="RVSUM_1:th.86">RVSUM_1:86</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a>, <a class="txt" href="matrprob.html#E3:56_2_2"><i><font color="Green" title="E62">A15</font></i></a>, <a class="txt" href="matrprob.html#E2:56"><i><font color="Green" title="E49">A2</font></i></a>, <a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="ref" href="matrprob.html#D4" target="_self" title="MATRPROB:def.4">Def4</a>, <a class="txt" href="matrprob.html#E1:56"><i><font color="Green" title="E48">A1</font></i></a>, <a class="txt" href="matrprob.html#E7:56_2_2"><i><font color="Green" title="E64">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">j</font>
<b>by </b><i><a class="txt" href="matrprob.html#E3:56_2_2"><i><font color="Green" title="E62">A15</font></i></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E13:56_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="rvsum_1.html#K10" title="RVSUM_1:func.10">*</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c5">i</font></span>)</span>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>,<font color="Maroon" title="c5">i</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E11:56_2"><i><font color="Green" title="E60">A13</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">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#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p2">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="matrprob.html#E9:56_2"><i><font color="Green" title="E59">A12</font></i></a>, <a class="ref" href="matrprob.html#T20" target="_self" title="MATRPROB:th.20">Th20</a>, <a class="txt" href="matrprob.html#E3:56"><i><font color="Green" title="E50">A3</font></i></a>, <a class="txt" href="matrprob.html#E2:56_2"><i><font color="Green" title="E52">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:56"/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixr1.html#K11" title="MATRIXR1:func.11">*</a> <font color="Maroon" title="c2">y</font></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <span class="p1">(<span class="default"><a href="matrprob.html#K6" title="MATRPROB:func.6">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c2">y</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a>, <a class="txt" href="matrprob.html#E5:56"><i><font color="Green" title="E51">A4</font></i></a></i>;<br/>


</div>
