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<b>let </b><font color="Maroon" title="c1">M</font> be  non <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a> <a href="matrprob.html#V5" title="MATRPROB:attr.5">with_line_sum=1</a> <a href="matrix_1.html#NM1" title="MATRIX_1:NM.1">Matrix</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>set </b><font color="Maroon" title="c2">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:84"/><i><font color="Green" title="E64">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="matrprob.html#T54" target="_self" title="MATRPROB:th.54">Th54</a></i>;<br/>
<a NAME="E2:84"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</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> <font color="Maroon" title="c1">M</font> &amp; ( for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c1">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</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> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="matrprob.html#D1" target="_self" title="MATRPROB:def.1">Def1</a></i>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:84_1_1"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c1">M</font></span>)</span>
 ;<br/><a NAME="E2:84_1_1"/><b>then </b>
<font color="Maroon" title="c3">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="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:84_1_1"/><b>then </b>
<font color="Maroon" title="c3">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> <font color="Maroon" title="c1">M</font></span>)</span>
 <b>by </b><i><a class="ref" href="matrprob.html#D1" target="_self" title="MATRPROB:def.1">Def1</a></i>;<br/><a NAME="E4:84_1_1"/><b>then </b>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">M</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E5:84_1_1"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="matrprob.html#D9" target="_self" title="MATRPROB:def.9">Def9</a></i>;<br/><b>hence </b><a NAME="E6:84_1_1"/>
<span class="p1">(<span class="default"><a href="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="matrprob.html#E1:84_1_1"><i><font color="Green" title="E65">A2</font></i></a>, <a class="ref" href="matrprob.html#D1" target="_self" title="MATRPROB:def.1">Def1</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" title="MATRPROB:func.3">Sum</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:84"><i><font color="Green" title="E64">A1</font></i></a>, <a class="ref" href="matrprob.html#T1" target="_self" title="MATRPROB:th.1">Th1</a></i>;<br/>
<b>hence </b> <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_2.html#K24" title="BINOP_2:func.24">*</a> 1
<b>by </b><i><a class="ref" href="rvsum_1.html#T110" title="RVSUM_1:th.110">RVSUM_1:110</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>

;<br/><br/>


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