<?xml version="1.0"?>
<div class="add">

<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/>
<b>set </b><font color="Maroon" title="c3">m</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:86_2_1"/><i><font color="Green" title="E64">A8</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>  &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</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="E2:86_2_1"/><i><font color="Green" title="E65">A9</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="matrprob.html#D6" target="_self" title="MATRPROB:def.6">Def6</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">e</font> =  <a href="matrprob.html#K4" title="MATRPROB:func.4">ColSum</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E3:86_2_1"/><i><font color="Green" title="E66">A10</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K4" title="MATRPROB:func.4">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> &amp; ( for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</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> holds <br/><span class="p1">(<span class="default"><a href="matrprob.html#K4" title="MATRPROB:func.4">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">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="Olive" title="b1">k</font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="matrprob.html#D2" target="_self" title="MATRPROB:def.2">Def2</a></i>;<br/>
<a NAME="E4:86_2_1"/><i><font color="Green" title="E67">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#K4" title="MATRPROB:func.4">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="matrprob.html#D2" target="_self" title="MATRPROB:def.2">Def2</a></i>;<br/>
<a NAME="E5:86_2_1"/><i><font color="Green" title="E68">A12</font></i>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#K4" title="MATRPROB:func.4">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrprob.html#K4" title="MATRPROB:func.4">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="86_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:86_2_1_1"/><i><font color="Green" title="E69">A13</font></i>: 
<font color="Maroon" title="c5">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#K4" title="MATRPROB:func.4">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span>
 ;<br/>

<a NAME="E2:86_2_1_1"/><i><font color="Green" title="E70">A14</font></i>: 
<font color="Maroon" title="c5">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="matrprob.html#E3:86_2_1"><i><font color="Green" title="E66">A10</font></i></a>, <a class="txt" href="matrprob.html#E1:86_2_1_1"><i><font color="Green" title="E69">A13</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:86_2_1_1"/><b>then </b>
for <font color="Olive" title="b1">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">j</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="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c1">M</font>,<font color="Maroon" title="c5">i</font></span>)</span> holds <br/><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="c5">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:86_2_1"><i><font color="Green" title="E65">A9</font></i></a>, <a class="txt" href="matrprob.html#E22"><i><font color="Green" title="E22">Lm3</font></i></a></i>;<br/>
<a NAME="E4:86_2_1_1"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">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="c5">i</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</a></i>;<br/>
<b>hence </b><a NAME="E5:86_2_1_1"/>
<span class="p1">(<span class="default"><a href="matrprob.html#K4" title="MATRPROB:func.4">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E2:86_2_1_1"><i><font color="Green" title="E70">A14</font></i></a>, <a class="ref" href="matrprob.html#D2" target="_self" title="MATRPROB:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="matrprob.html#K4" title="MATRPROB:func.4">ColSum</a> <font color="Maroon" title="c1">M</font></span>)</span> = 
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">M</font>
<b>by </b><i><a class="ref" href="matrprob.html#T29" target="_self" title="MATRPROB:th.29">Th29</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrprob.html#D7" target="_self" title="MATRPROB:def.7">Def7</a></i>
;<br/>
<b>hence </b><a NAME="E7:86_2_1"/>
( not  <a href="matrprob.html#NK10" title="MATRPROB:NK.10">Column_Marginal</a> <font color="Maroon" title="c1">M</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="matrprob.html#NK10" title="MATRPROB:NK.10">Column_Marginal</a> <font color="Maroon" title="c1">M</font> is <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a> )
 <b>by </b><i><a class="txt" href="matrprob.html#E1:86_2_1"><i><font color="Green" title="E64">A8</font></i></a>, <a class="txt" href="matrprob.html#E4:86_2_1"><i><font color="Green" title="E67">A11</font></i></a>, <a class="txt" href="matrprob.html#E5:86_2_1"><i><font color="Green" title="E68">A12</font></i></a>, <a class="ref" href="matrprob.html#D5" target="_self" title="MATRPROB:def.5">Def5</a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>


</div>
