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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="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span><br/>

<a NAME="E1:64"/><span class="lab"><font color="Green" title="E45">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</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">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></span>;<br/>
<a NAME="E2:64"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</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">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E47">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">i</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="Red">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <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"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> implies <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c2">i</font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E48">A4</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">and </span><br/><a NAME="E2:64_1"/><span class="lab"><font color="Green" title="E49">A5</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"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c2">i</font></span><br/><a NAME="E3:64_1"/>
<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"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/><a NAME="E4:64_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><a NAME="E5:64_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A6</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></span>;<br/><a NAME="E6:64_1"/><span class="lab"><font color="Green" title="E51">A7</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c2">i</font> = 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c2">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></span>
<br/>.= 
 <a href="rvsum_1.html#K17" title="RVSUM_1:func.17">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c2">i</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="matrix_2.html#T17" title="MATRIX_2:th.17">MATRIX_2:17</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="matrix_5.html#K9" title="MATRIX_5:func.9">.</a> <font color="Maroon" title="c2">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:64_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:64"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</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> <span class="p1">(<span class="default"><a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:64"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:64"/>
 <a href="matrixc1.html#K11" title="MATRIXC1:func.11">ColSum</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixc1.html#K10" title="MATRIXC1:func.10">LineSum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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