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<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#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">M</font> being  <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st  <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="Olive" title="b1">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="Olive" title="b1">M</font> holds <br/><span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">M</font>,<font color="Maroon" title="c2">y</font></span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span><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#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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> implies <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> )</span><br/>



<b>assume </b><a NAME="E1:65"/><i><font color="Green" title="E49">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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span><br/>

<a NAME="E2:65"/><i><font color="Green" title="E50">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </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="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E3:65"/>
 <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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </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="c3">M</font>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E4:65"/><b>then </b><i><font color="Green" title="E51">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </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"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>;<br/>
<a NAME="E5:65"/><i><font color="Green" title="E52">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrixc1.html#E2:65"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E6:65"/><i><font color="Green" title="E53">A5</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"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </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="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E7:65"/><i><font color="Green" title="E54">A6</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<i><font color="Green" title="E55">A7</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"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
<br/>.= 
 <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="txt" href="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>
;<br/>
<i><font color="Green" title="E56">A8</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:65"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="matrixc1.html#E5:65"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>
<br/>.= 
 <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="txt" href="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
;<br/>
<i><font color="Green" title="E57">A9</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> = 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
<br/>.= 
 <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="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>
;<br/>
<i><font color="Green" title="E58">A10</font></i>:  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:65"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="matrixc1.html#E5:65"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>
<br/>.= 
 <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="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
;<br/>
<a NAME="E12:65"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> )</span><br/>



<b>reconsider </b><font color="Maroon" title="c6">i'</font> = <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c7">j'</font> = <font color="Maroon" title="c5">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:65_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span><br/>

<a NAME="E3:65_5"/><b>then </b><i><font color="Green" title="E59">A11</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">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="matrixc1.html#K13" title="MATRIXC1:func.13">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="txt" href="matrixc1.html#E6:65"><i><font color="Green" title="E53">A5</font></i></a>, <a class="txt" href="matrixc1.html#E7:65"><i><font color="Green" title="E54">A6</font></i></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<a NAME="E4:65_5"/><b>then </b><i><font color="Green" title="E60">A12</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<a NAME="E5:65_5"/><i><font color="Green" title="E61">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">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="c1">x</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">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> <font color="Maroon" title="c2">y</font> )
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrixc1.html#E3:65_5"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>;<br/>
<a NAME="E6:65_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">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="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrixc1.html#E3:65_5"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>;<br/>
<a NAME="E7:65_5"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">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"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>;<br/>
<a NAME="E8:65_5"/><b>then </b><i><font color="Green" title="E62">A14</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<a NAME="E9:65_5"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c4">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> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j</font> &amp; <font color="Maroon" title="c5">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="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrixc1.html#E3:65_5"><i><font color="Green" title="E59">A11</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>;<br/>
<a NAME="E10:65_5"/><b>then </b><i><font color="Green" title="E63">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">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="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></i>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="txt" href="matrixc1.html#E4:65_5"><i><font color="Green" title="E60">A12</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<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="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:65"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="matrixc1.html#E10:65_5"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<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="c6">i'</font>,<font color="Maroon" title="c7">j'</font></span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<b>by </b><i><a class="txt" href="matrixc1.html#E5:65_5"><i><font color="Green" title="E61">A13</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<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="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T121" title="COMPLEX1:th.121">COMPLEX1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p3">(<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="c6">i'</font>,<font color="Maroon" title="c7">j'</font></span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:65"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrixc1.html#E5:65_5"><i><font color="Green" title="E61">A13</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<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="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="ref" href="complex1.html#T121" title="COMPLEX1:th.121">COMPLEX1:121</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E10:65_5"><i><font color="Green" title="E63">A15</font></i></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c7">j'</font></span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E2:65"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="matrixc1.html#E5:65_5"><i><font color="Green" title="E61">A13</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i'</font>,<font color="Maroon" title="c7">j'</font></span>)</span></span>)</span> <a href="binop_2.html#K5" title="BINOP_2:func.5">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> </span>)</span>
<b>by </b><i><a class="txt" href="matrixc1.html#E5:65_5"><i><font color="Green" title="E61">A13</font></i></a>, <a class="ref" href="complsp2.html#D1" title="COMPLSP2:def.1">COMPLSP2:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:65"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="matrixc1.html#E5:65"><i><font color="Green" title="E52">A4</font></i></a>, <a class="txt" href="matrixc1.html#E8:65_5"><i><font color="Green" title="E62">A14</font></i></a>, <a class="ref" href="matrixc1.html#D13" target="_self" title="MATRIXC1:def.13">Def13</a></i>
;<br/>
<b>hence </b><a NAME="E12:65_5"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E13:65"/>
<span class="p1">(<span class="default"><a href="matrixc1.html#K13" title="MATRIXC1:func.13">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="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixc1.html#K13" title="MATRIXC1:func.13">QuadraticForm</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">x</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="complsp2.html#K1" title="COMPLSP2:func.1">*'</a> </span>)</span>
 <b>by </b><i><a class="txt" href="matrixc1.html#E8:65"><i><font color="Green" title="E55">A7</font></i></a>, <a class="txt" href="matrixc1.html#E9:65"><i><font color="Green" title="E56">A8</font></i></a>, <a class="txt" href="matrixc1.html#E10:65"><i><font color="Green" title="E57">A9</font></i></a>, <a class="txt" href="matrixc1.html#E11:65"><i><font color="Green" title="E58">A10</font></i></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
