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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#V4" title="MATRPROB:attr.4">Joint_Probability</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/>
<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="E1:75"/><i><font color="Green" title="E60">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> 0 &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> 0 )
 
<b>by </b><i><a class="ref" href="matrprob.html#T54" target="_self" title="MATRPROB:th.54">Th54</a></i>;<br/>
<a NAME="E2:75"/><b>then </b><i><font color="Green" title="E61">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">M1</font> = <font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">M</font>, <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">M</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></i>;<br/>
<a NAME="E4:75"/>
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="c4">M1</font> holds <br/><font color="Maroon" title="c4">M1</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> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</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="E1:75_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">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="c4">M1</font>
 ;<br/>

<a NAME="E2:75_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</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>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E3:75_1"/><b>then </b>
( <font color="Maroon" title="c4">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">j</font>,<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 )
 
<b>by </b><i><a class="ref" href="matrprob.html#D6" target="_self" title="MATRPROB:def.6">Def6</a>, <a class="ref" href="matrix_1.html#D7" title="MATRIX_1:def.7">MATRIX_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1"/>
<font color="Maroon" title="c4">M1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:75"/><b>then </b><i><font color="Green" title="E62">A3</font></i>: 
<font color="Maroon" title="c4">M1</font> is <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a>
 
<b>by </b><i><a class="ref" href="matrprob.html#D6" target="_self" title="MATRPROB:def.6">Def6</a></i>;<br/>
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c4">M1</font> = 
 <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#T28" target="_self" title="MATRPROB:th.28">Th28</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/>
<a NAME="E7:75"/><b>then </b>
<font color="Maroon" title="c4">M1</font> is <a href="matrprob.html#V3" title="MATRPROB:attr.3">with_sum=1</a>
 
<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="E8:75"/>
<font color="Maroon" title="c1">M</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a>  is  non <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a> <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_Probability</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> 
 <b>by </b><i><a class="ref" href="matrprob.html#D8" target="_self" title="MATRPROB:def.8">Def8</a>, <a class="txt" href="matrprob.html#E1:75"><i><font color="Green" title="E60">A1</font></i></a>, <a class="txt" href="matrprob.html#E2:75"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a>, <a class="txt" href="matrprob.html#E5:75"><i><font color="Green" title="E62">A3</font></i></a></i>;<br/>


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