<?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>assume </b><a NAME="E1:55"/><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="E2:55"/><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">y</font>
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E3:55"/><i><font color="Green">E49</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<i><font color="Green">E50</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="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="matrix_1.html#D7">MATRIX_1:def 7</a></i>
<br/>.= 
 <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>, </i>
;<br/>
<a NAME="E5:55"/><i><font color="Green">E51</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="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="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E6:55"/><i><font color="Green">E58</font></i>: 
 <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">y</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">x</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#E2:55"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrprob.html#E3:55"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E7:55"/><i><font color="Green">E59</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>, </i>;<br/>
<a NAME="E8:55"/><i><font color="Green">E60</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><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> <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="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="matrix_2.html#T12">MATRIX_2:12</a>, , </i>;<br/>
<i><font color="Green">E61</font></i>:  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="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="matrix_2.html#T12">MATRIX_2:12</a>, , </i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, </i>
;<br/>
<a NAME="E10:55"/><i><font color="Green">E62</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">y</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">x</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="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E2:55"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrprob.html#E3:55"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E11:55"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">j</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:55_3"/><i><font color="Green">E63</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font>, <font color="Maroon">j</font> = <font color="Maroon">j</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E3:55_3"/><i><font color="Green">E77</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="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> &amp; 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="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="matrprob.html#T3" target="_self">Th3</a>, <a class="txt" href="matrprob.html#E5:55"><i><font color="Green">E51</font></i></a>, , <a class="ref" href="matrixc1.html#T1">MATRIXC1:1</a></i>;<br/>
<a NAME="E4:55_3"/><i><font color="Green">E117</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="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp; 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> <font color="Maroon">M</font> )
 
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E6:55"><i><font color="Green">E58</font></i></a>, </i>;<br/>
<a NAME="E5:55_3"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</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="txt" href="matrprob.html#E7:55"><i><font color="Green">E59</font></i></a>, <a class="ref" href="matrixc1.html#T1">MATRIXC1:1</a></i>;<br/>
<a NAME="E6:55_3"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E6:55"><i><font color="Green">E58</font></i></a>, , <a class="txt" href="matrprob.html#E2:55"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrprob.html#E3:55"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E7:55_3"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1">MATRIXC1:1</a></i>;<br/>
<a NAME="E8:55_3"/><i><font color="Green">E120</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="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="txt" href="matrprob.html#E6:55"><i><font color="Green">E58</font></i></a>, <a class="ref" href="matrixc1.html#T1">MATRIXC1:1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</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"><span class="p3">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E2:55"><i><font color="Green">E48</font></i></a>, <a class="txt" href="matrprob.html#E3:55"><i><font color="Green">E49</font></i></a>, <a class="txt" href="matrprob.html#E9:55"><i><font color="Green">E61</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</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">j</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="binop_2.html#K11">*</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="txt" href="matrprob.html#E8:55"><i><font color="Green">E60</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">j</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">j</font>,<font color="Maroon">i</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">i</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">j</font>,<font color="Maroon">i</font>
<b>by </b><i><a class="ref" href="matrprob.html#T1" target="_self">Th1</a>, <a class="txt" href="matrprob.html#E8:55"><i><font color="Green">E60</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a>, <a class="txt" href="matrprob.html#E10:55"><i><font color="Green">E62</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E10:55_3"/>
<span class="p1">(<span class="default"><a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">y</font>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><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> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E12:55"/>
<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#K4">@</a>  <a href="hidden.html#R1">=</a>  <a href="matrprob.html#K6" target="_self">QuadraticForm</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#T21">MATRIX_1:21</a>, <a class="txt" href="matrprob.html#E4:55"><i><font color="Green">E50</font></i></a>, <a class="ref" href="matrprob.html#T2" target="_self">Th2</a>, , </i>;<br/>


</div>
