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<b>let </b><font color="Maroon">P1</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> , <font color="Maroon">P2</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="E1:90"/><i><font color="Green">E43</font></i>: 
 <a href="matrix_1.html#K1">width</a> <font color="Maroon">P1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font>
 ;<br/>

<b>set </b><font color="Maroon">n1</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font>;<br/>
<b>set </b><font color="Maroon">m</font> =  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font>;<br/>
<b>set </b><font color="Maroon">n2</font> =  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P2</font>;<br/>
<a NAME="E2:90"/><i><font color="Green">E48</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P2</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E3:90"/><i><font color="Green">E49</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P2</font> &amp; ( 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#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">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</font></span>)</span></span>)</span> holds <br/> <a href="matrix_1.html#K8">Line</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</font></span>)</span>,<font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8">Line</a> <font color="Maroon">P1</font>,<font color="Olive">i</font></span>)</span> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P2</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>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</font> as    <a href="matrix_1.html#M1">Matrix</a> of  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font>, <a href="matrix_1.html#K1">width</a> <font color="Maroon">P2</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="E5:90"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">P</font> is  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> 
 
<b>by </b><i><a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, , <a class="ref" href="goboard1.html#D5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E6:90"/>
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/> <a href="matrix_1.html#K8">Line</a> <font color="Maroon">P</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="90_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>assume </b><a NAME="E1:90_1"/><i><font color="Green">E51</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">P</font>
 ;<br/>

<a NAME="E2:90_1"/><i><font color="Green">E58</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> <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="E3:90_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">P1</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">p</font> =  <a href="matrix_1.html#K8">Line</a> <font color="Maroon">P1</font>,<font color="Maroon">i</font> as  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/>;<br/>
<a NAME="E5:90_1"/>
 <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">P2</font>
 
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="matrix_1.html#D8">MATRIX_1:def 8</a></i>;<br/>
<a NAME="E6:90_1"/><b>then </b>
( <font color="Maroon">p</font> <a href="matrixr1.html#K12">*</a> <font color="Maroon">P2</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">P2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P2</font> )
 
<b>by </b><i><a class="ref" href="matrprob.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E7:90_1"/>
 <a href="matrix_1.html#K8">Line</a> <font color="Maroon">P</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#T5" target="_self">Th5</a>, <a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:90"/>
( <font color="Maroon">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</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>  &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">P1</font> &amp;  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="matrixr1.html#K6">*</a> <font color="Maroon">P2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <font color="Maroon">P2</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>, , , </i>;<br/>


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