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<span class="kw">let </span><font color="Maroon" title="c1">M1</font>, <font color="Maroon" title="c2">M2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M1</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="c2">M2</font> &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M1</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="c2">M2</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:9"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M1</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="c2">M2</font>
 <span class="kw">and </span><br/><a NAME="E2:9"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M1</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="c2">M2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span></span><br/>

<a NAME="E3:9"/><span class="lab"><font color="Green" title="E11">A3</font></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">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#T6" target="_self" title="MATRIXC1:th.6">Th6</a></span>;<br/>
<a NAME="E4:9"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
 <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"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/>
<a NAME="E5:9"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></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> <font color="Maroon" title="c1">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#T6" target="_self" title="MATRIXC1:th.6">Th6</a></span>;<br/>
<a NAME="E6:9"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 <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">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E15">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:9_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span><br/><a NAME="E2:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E3:9_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E4:9_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c4">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="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E5:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c4">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="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/><a NAME="E6:9_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c3">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="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E7:9_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">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="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/><a NAME="E8:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E3:9_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E5:9_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E9:9_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E10:9_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c1">M1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E6:9_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E4:9_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E11:9_1"/><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c2">M2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:9"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:9"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:9_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E6:9_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E4:9_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E12:9_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">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="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:9_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E6:9_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E2:9_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E4:9_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E13:9_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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="c3">i</font>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</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="c3">i</font>,<font color="Maroon" title="c4">j</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="matrixc1.html#T7" target="_self" title="MATRIXC1:th.7">Th7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T32" title="COMPLEX1:th.32">COMPLEX1:32</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</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="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:9_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</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="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="binop_2.html#K3" title="BINOP_2:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</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="c3">i</font>,<font color="Maroon" title="c4">j</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:9_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:9_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="matrixc1.html#T7" target="_self" title="MATRIXC1:th.7">Th7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E8:9"/><span class="lab"><font color="Green" title="E16">A18</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/>
<a NAME="E9:9"/><span class="lab"><font color="Green" title="E17">A19</font></span>: 
 <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"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span></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="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#T6" target="_self" title="MATRIXC1:th.6">Th6</a></span>;<br/>
<a NAME="E10:9"/><span class="lab"><font color="Green" title="E18">A20</font></span>: 
 <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">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#T6" target="_self" title="MATRIXC1:th.6">Th6</a></span>;<br/>
<a NAME="E11:9"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</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="c1">M1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <font color="Maroon" title="c2">M2</font></span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M1</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrix_5.html#K3" title="MATRIX_5:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M2</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:9"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="txt" href="#E3:9"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E10:9"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E4:9"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E5:9"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E9:9"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E8:9"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E7:9"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="matrix_1.html#T21" title="MATRIX_1:th.21">MATRIX_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
