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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#V4" target="_self">Joint_Probability</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/>
<b>set </b><font color="Maroon">m</font> =  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>;<br/>
<a NAME="E1:75"/><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 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E2:75"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a> </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><a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">M1</font> = <font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a>  as    <a href="matrix_1.html#M1">Matrix</a> of  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>, <a href="finseq_1.html#K3">len</a> <font color="Maroon">M</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="matrix_2.html#T7">MATRIX_2:7</a></i>;<br/>
<a NAME="E4:75"/>
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font> holds <br/><font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><a NAME="E1:75_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M1</font>
 ;<br/>

<a NAME="E2:75_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">j</font>,<font color="Maroon">i</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<a NAME="E3:75_1"/><b>then </b>
( <font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> &amp; <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">j</font>,<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="matrix_1.html#D7">MATRIX_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1"/>
<font color="Maroon">M1</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:75"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">M1</font> is <a href="matrprob.html#V2" target="_self">m-nonnegative</a>
 
<b>by </b><i/>;<br/>
 <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M1</font> = 
 <a href="matrprob.html#K5" target="_self">SumAll</a> <font color="Maroon">M</font>
<b>by </b><i/>
<br/>.= 
1
<b>by </b><i/>
;<br/>
<a NAME="E7:75"/><b>then </b>
<font color="Maroon">M1</font> is <a href="matrprob.html#V3" target="_self">with_sum=1</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:75"/>
<font color="Maroon">M</font> <a href="matrix_1.html#K4">@</a>  is  non <a href="relat_1.html#V3">empty-yielding</a> <a href="matrprob.html#V4" target="_self">Joint_Probability</a> <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i><a class="ref" href="matrprob.html#T3" target="_self">Th3</a>, , , <a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a>, </i>;<br/>


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