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<div class="add">

<b>let </b><font color="Maroon">M</font> be  non <a href="relat_1.html#V3">empty-yielding</a> <a href="matrprob.html#V5" target="_self">with_line_sum=1</a> <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>set </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>;<br/>
<a NAME="E1:84"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E2:84"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> &amp; ( for <font color="Olive">i</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> 1 ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:84_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>
 <b>by </b><i/>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span>
 ;<br/><a NAME="E2:84_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:84_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E4:84_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:84_1_1"/><b>then </b>
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:84_1_1"/>
<span class="p1">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, </i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:84"/><b>then </b>
 <a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1
 
<b>by </b><i><a class="ref" href="matrprob.html#T4" target="_self">Th4</a>, </i>;<br/>
<b>hence </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font></span>)</span> <a href="binop_2.html#K24">*</a> 1
<b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>

;<br/><br/>


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