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<div class="add">

<b>let </b><font color="Maroon">p1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> , <font color="Maroon">p2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K2">COMPLEX</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E2:59_1_2"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span> holds <br/><font color="Maroon">p1</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">j</font></span>)</span>
 <b>and </b><br/><a NAME="E3:59_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 <b>and </b><br/><a NAME="E4:59_1_2"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span> holds <br/><font color="Maroon">p2</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Olive">j</font></span>)</span>
 ;<br/>

<a NAME="E5:59_1_2"/>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span> holds <br/><font color="Maroon">p1</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="matrix_5.html#K9">.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:59_1_2_1"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font></span>)</span>
 ;<br/>

<a NAME="E2:59_1_2_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">p2</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="matrixc1.html#K6" target="_self">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">M</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrixc1.html#T3" target="_self">Th3</a>, <a class="ref" href="matrixc1.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:59_1_2_1"/>
<font color="Maroon">p1</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="matrix_5.html#K9">.</a> <font color="Maroon">j</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:59_1_2"/>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="matrixc1.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


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