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<b>let </b><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> ;<br/>

<a NAME="E1:61"/><i><font color="Green" title="E45">A1</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#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>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>;<br/>
<a NAME="E2:61"/><i><font color="Green" title="E46">A2</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#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>
 
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>;<br/>
<a NAME="E3:61"/><b>then </b><i><font color="Green" title="E47">A3</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#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>
 
<b>by </b><i><a class="txt" href="matrixc1.html#E1:61"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:61_1"/><i><font color="Green" title="E48">A4</font></i>: 
( 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> )
 ;<br/><a NAME="E2:61_1"/><b>then </b>
( 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#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> )
 <b>by </b><i><a class="txt" href="matrixc1.html#E1:61"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="matrixc1.html#E2:61"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/><a NAME="E3:61_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E4:61_1"/><b>then </b><i><font color="Green" title="E49">A5</font></i>: 
<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>
 <b>by </b><i><a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>;<br/><a NAME="E5:61_1"/><i><font color="Green" title="E50">A6</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="matrixc1.html#E2:61"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="matrixc1.html#E1:61_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><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>
<b>by </b><i><a class="ref" href="matrixc1.html#D11" target="_self" title="MATRIXC1:def.11">Def11</a></i>
<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>
<b>by </b><i><a class="txt" href="matrixc1.html#E5:61_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="matrix_2.html#T17" title="MATRIX_2:th.17">MATRIX_2:17</a></i>
<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>
<b>by </b><i><a class="txt" href="matrixc1.html#E4:61_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="matrixc1.html#D10" target="_self" title="MATRIXC1:def.10">Def10</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:61"/>
 <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>
 <b>by </b><i><a class="txt" href="matrixc1.html#E3:61"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></i>;<br/>


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