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

<b>let </b><font color="Maroon">p</font> be  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/><b>let </b><font color="Maroon">P</font> be  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/>



<b>assume </b><a NAME="E1:89"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P</font>
 ;<br/>

<b>set </b><font color="Maroon">n</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font>;<br/>
<a NAME="E2:89"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E3:89"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="matrixr1.html#T62">MATRIXR1:62</a></i>;<br/>
<b>set </b><font color="Maroon">e</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1;<br/>
<a NAME="E4:89"/><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E5:89"/><i><font color="Green">E51</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P</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">P</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">P</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">P</font> )
 
<b>by </b><i><a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, <a class="ref" href="matrix_2.html#T12">MATRIX_2:12</a></i>;<br/>
<a NAME="E6:89"/><b>then </b><i><font color="Green">E58</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, , <a class="ref" href="matrixr1.html#T62">MATRIXR1:62</a></i>;<br/>
<a NAME="E7:89"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1"><b>proof </b></a><div class="add">

<a NAME="E1:89_1"/>
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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_1_1"><b>proof </b></a><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="E1:89_1_1"/><i><font color="Green">E60</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:89_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:89_1_1"/><b>then </b><i><font color="Green">E62</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">P</font>
 
<b>by </b><i>, <a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span></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"><a href="matrix_1.html#K9">Col</a> <span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, , , <a class="txt" href="matrprob.html#E5:89"><i><font color="Green">E51</font></i></a></i>
<br/>.= 
 <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">P</font>,<font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_2.html#T16">MATRIX_2:16</a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrix_2.html#T18">MATRIX_2:18</a></i>
<br/>.= 
1
<b>by </b><i>, <a class="txt" href="matrprob.html#E1:89_1_1"><i><font color="Green">E60</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:89_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1
 <b>by </b><i><a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, , </i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E63</font></i>:  <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span> = 
<span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K1">|(</a><span class="default"><font color="Maroon">p</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span> <a href="newton.html#K1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12">*</a> <span class="p3">(<span class="default"><font color="Maroon">P</font> <a href="matrix_1.html#K4">@</a> </span>)</span></span>)</span></span><a href="euclid_2.html#K1">)|</a></span>
<b>by </b><i><a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, <a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, , <a class="txt" href="matrprob.html#E4:89"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">p</font>
<b>by </b><i>, </i>
<br/>.= 
1
<b>by </b><i/>
;<br/>
<a NAME="E9:89"/>
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> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_3"><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>assume </b><a NAME="E1:89_3"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span>
 ;<br/>

<a NAME="E2:89_3"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:89_3"/><i><font color="Green">E118</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">p</font> holds <br/><font color="Maroon">p</font> <a href="seq_1.html#K2">.</a> <font color="Olive">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E4:89_3"/><i><font color="Green">E119</font></i>: 
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">P</font> holds <br/><font color="Maroon">P</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="E5:89_3"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, </i>;<br/>
<a NAME="E6:89_3"/><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#K9">Col</a> <font color="Maroon">P</font>,<font color="Maroon">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">P</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>, </i>;<br/>
<a NAME="E7:89_3"/><b>then </b><i><font color="Green">E120</font></i>: 
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> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">P</font>,<font color="Maroon">i</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">P</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="matrprob.html#T7" target="_self">Th7</a>, </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> = 
<font color="Maroon">p</font> <a href="matrprob.html#NK7" target="_self">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">P</font>,<font color="Maroon">i</font></span>)</span>
<b>by </b><i>, <a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="txt" href="matrprob.html#E1:89"><i><font color="Green">E43</font></i></a></i>
<br/>.= 
 <a href="rvsum_1.html#K15">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K13">mlt</a> <font color="Maroon">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9">Col</a> <font color="Maroon">P</font>,<font color="Maroon">i</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E9:89_3"/>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="matrprob.html#E1:89_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="rvsum_1.html#T114">RVSUM_1:114</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:89"/>
( <font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</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>  &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P</font> )
 <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>
