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<b>let </b><font color="Maroon">m</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">M</font> be    <a href="matrix_1.html#M1">Matrix</a> of <font color="Maroon">m</font>,0, <a href="numbers.html#K1">REAL</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1"/>
( <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><b>suppose </b></a><a NAME="E1:33_1_1"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:33_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1"/>
 <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2"><b>suppose </b></a><a NAME="E1:33_1_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:33_1_2"/>
(  <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="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; ( for <font color="Olive">k</font> being  <a href="nat_1.html#NM1">Nat</a>  st <font color="Olive">k</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">k</font> <a href="hidden.html#R1">=</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_2_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</a>  <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>
 
<b>by </b><i/>;<br/>
<a NAME="E2:33_1_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> <a href="hidden.html#R1">=</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>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E3:33_1_2_1"/>
 <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>, <a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_2_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="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i/>;<br/>

<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon">k</font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:33_1_2_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</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="E2:33_1_2_1_1"/>
<font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">M</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:33_1_2_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="matrix_1.html#D3">MATRIX_1:def 3</a></i>;<br/><b>thus </b><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">k</font> = 
 <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">k</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>
;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div><b>hence </b> <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font> = 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="matrprob.html#K3" target="_self">Sum</a> <font color="Maroon">M</font></span>)</span></span>)</span> <a href="newton.html#K1">|-&gt;</a> 0</span>)</span>
<b>by </b><i/>
<br/>.= 
<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> <a href="binop_2.html#K24">*</a> 0
<b>by </b><i><a class="ref" href="rvsum_1.html#T110">RVSUM_1:110</a></i>
<br/>.= 
0

;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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