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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="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:83_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">M</font> is  non <a href="relat_1.html#V3">empty-yielding</a> <a href="matrprob.html#V6" target="_self">Conditional_Probability</a> <a href="matrix_1.html#NM1">Matrix</a> of <a href="numbers.html#K1">REAL</a> 
 ;<br/><div><b>hereby </b>
<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>assume </b><a NAME="E1:83_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 ;<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="E2:83_1_1"/><i><font color="Green">E49</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="E3:83_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">M</font>
 <b>by </b><i><a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/><i><font color="Green">E51</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> = 
 <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">i</font></span>)</span>
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="matrprob.html#T4" target="_self">Th4</a>, <a class="txt" href="matrprob.html#E5:83"><i><font color="Green">E60</font></i></a></i>
;<br/><a NAME="E5:83_1_1"/>
( ( 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">M</font> holds <br/><font color="Maroon">M</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 ) &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> holds <br/> <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="Olive">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1 ) )
 <b>by </b><i><a class="ref" href="matrprob.html#T4" target="_self">Th4</a>, , <a class="txt" href="matrprob.html#E5:83"><i><font color="Green">E60</font></i></a></i>;<br/><a NAME="E6:83_1_1"/><b>then </b>
for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, </i>;<br/><b>hence </b><a NAME="E7:83_1_1"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="matrprob.html#V1" target="_self">ProbFinS</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 <b>by </b><i>, , , <a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/></div>
<b>end;</b></div></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:83"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> holds <br/> <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Olive">i</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="matrprob.html#V1" target="_self">ProbFinS</a>  <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<a NAME="E3:83"/>
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 <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Olive">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Olive">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <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="83_2"><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="E1:83_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> )
 ;<br/>

<a NAME="E2:83_2"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font> is <a href="matrprob.html#V1" target="_self">ProbFinS</a>
 
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="matrprob.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:83_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:83"/><b>then </b>
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">M</font> holds <br/><font color="Maroon">M</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
 
<b>by </b><i/>;<br/>
<a NAME="E5:83"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">M</font> is <a href="matrprob.html#V2" target="_self">m-nonnegative</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:83_3"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">M</font>
 ;<br/><a NAME="E2:83_3"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font> is <a href="matrprob.html#V1" target="_self">ProbFinS</a>
 <b>by </b><i><a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, <a class="ref" href="matrprob.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:83_3"/><b>then </b><i><font color="Green">E62</font></i>: 
( ( for <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp;  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">M</font>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:83_3"/>
 <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> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:83"/><b>then </b>
<font color="Maroon">M</font> is <a href="matrprob.html#V5" target="_self">with_line_sum=1</a>
 
<b>by </b><i><a class="txt" href="matrprob.html#E5:83"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:83"/>
<font color="Maroon">M</font> is  non <a href="relat_1.html#V3">empty-yielding</a> <a href="matrprob.html#V6" target="_self">Conditional_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#T5" target="_self">Th5</a>, </i>;<br/>


</div>
