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

<span class="kw">let </span><font color="Maroon" title="c1">P1</font>, <font color="Maroon" title="c2">P2</font> be  non  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>   <a href="matrprob.html#V6" title="MATRPROB:attr.6">Conditional_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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">P1</font> <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="c2">P2</font> implies ( <font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font> is  non  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>   <a href="matrprob.html#V6" title="MATRPROB:attr.6">Conditional_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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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">P1</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">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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="c2">P2</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:87"/><span class="lab"><font color="Green" title="E65">A1</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c1">P1</font> <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="c2">P2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font> is  non  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>   <a href="matrprob.html#V6" title="MATRPROB:attr.6">Conditional_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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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">P1</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">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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="c2">P2</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c3">n2</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">m</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">P2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">n1</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">P1</font>;<br/>
<a NAME="E2:87"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="matrprob.html#T42" target="_self" title="MATRPROB:th.42">Th42</a></span>;<br/>
<a NAME="E3:87"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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="c2">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="matrprob.html#T42" target="_self" title="MATRPROB:th.42">Th42</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">P</font> = <font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font> as    <a href="matrix_1.html#M1" title="MATRIX_1:mode.1">Matrix</a> of  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">P1</font>, <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P2</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="matrix_2.html#T7" title="MATRIX_2:th.7">MATRIX_2:7</a></span>;<br/>
<a NAME="E5:87"/><span class="lab"><font color="Green" title="E68">A4</font></span>: 
for <font color="Olive" title="b1">i</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 <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">P</font> holds <br/> <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">P</font>,<font color="Olive" title="b1">i</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">i</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">P</font> implies  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">i</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:87_1"/>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">i</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><br/>

<a NAME="E2:87_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A5</font></span>: 
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E3:87_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c1">P1</font>,<font color="Maroon" title="c7">i</font> as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#T60" target="_self" title="MATRPROB:th.60">Th60</a></span>;<br/>
<a NAME="E5:87_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <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="c2">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E6:87_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P2</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#T62" target="_self" title="MATRPROB:th.62">Th62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:87_1"/>
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c7">i</font> is  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E2:87_1"><span class="lab"><font color="Green" title="E69">A5</font></span></a>, <a class="ref" href="matrprob.html#T42" target="_self" title="MATRPROB:th.42">Th42</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E6:87"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">P1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#T54" target="_self" title="MATRPROB:th.54">Th54</a></span>;<br/>
<a NAME="E7:87"/><span class="kw">then </span>
<font color="Maroon" title="c6">P</font> is  non  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:87"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E3:87"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:87"/>
( <font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font> is  non  <a href="relat_1.html#V3" title="RELAT_1:attr.3">empty-yielding</a>   <a href="matrprob.html#V6" title="MATRPROB:attr.6">Conditional_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>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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">P1</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">P1</font> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">P2</font></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="c2">P2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:87"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E5:87"><span class="lab"><font color="Green" title="E68">A4</font></span></a>, <a class="ref" href="matrprob.html#T42" target="_self" title="MATRPROB:th.42">Th42</a>, <a class="ref" href="matrprob.html#T60" target="_self" title="MATRPROB:th.60">Th60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
