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<span class="kw">let </span><font color="Maroon" title="c1">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font></span><br/>

<a NAME="E1:3"/><span class="lab"><font color="Green" title="E4">A1</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">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="c1">M</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="E2:3"/><span class="lab"><font color="Green" title="E5">A2</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">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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">M</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#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/>
<a NAME="E3:3"/><span class="lab"><font color="Green" title="E6">A3</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">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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M</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="E4:3"/><span class="lab"><font color="Green" title="E7">A4</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">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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">M</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#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">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="c2">i</font>,<font color="Maroon" title="c3">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">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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="c2">i</font>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>) )</span><br/><span class="kw">assume </span><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">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">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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="c2">i</font>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>)</span><br/><a NAME="E2:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A6</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</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="E3:3_1"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E4:3_1"/><span class="lab"><font color="Green" title="E11">A8</font></span>: 
<font color="Maroon" title="c3">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">M</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="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E5:3_1"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<font color="Maroon" title="c2">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">M</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="#E4:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E6:3_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">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">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</a></span>;<br/><a NAME="E7:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c2">i</font>,<font color="Maroon" title="c3">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">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/><a NAME="E8:3_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">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">M</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="#E2:3_1"><span class="lab"><font color="Green" title="E9">A6</font></span></a>, <a class="txt" href="#E5:3_1"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E3:3_1"><span class="lab"><font color="Green" title="E10">A7</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E11">A8</font></span></a>, <a class="ref" href="matrixc1.html#T1" target="_self" title="MATRIXC1:th.1">Th1</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">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></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="c2">i</font>,<font color="Maroon" title="c3">j</font>) = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>)</span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a></span>)</span> <a href="complex1.html#K15" title="COMPLEX1:func.15">*'</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:3_1"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="ref" href="matrixc1.html#D1" target="_self" title="MATRIXC1:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>)

;<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>
<span class="kw">hence </span><a NAME="E6:3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a></span>)</span> <a href="matrixc1.html#K1" title="MATRIXC1:func.1">*'</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:3"><span class="lab"><font color="Green" title="E7">A4</font></span></a>, <a class="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:3"><span class="lab"><font color="Green" title="E5">A2</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E6">A3</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/>


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