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

<b>let </b><font color="Maroon" title="c1">p</font> be  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> ;<br/><b>let </b><font color="Maroon" title="c2">P</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> ;<br/>



<b>assume </b><a NAME="E1:89"/><i><font color="Green" title="E65">A1</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">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">P</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">n</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font>;<br/>
<b>set </b><font color="Maroon" title="c4">m</font> =  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font>;<br/>
<a NAME="E2:89"/><i><font color="Green" title="E66">A2</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</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">P</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>  )
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:89"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="matrprob.html#T54" target="_self" title="MATRPROB:th.54">Th54</a></i>;<br/>
<a NAME="E3:89"/><i><font color="Green" title="E67">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</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">P</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:89"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="matrixr1.html#T62" title="MATRIXR1:th.62">MATRIXR1:62</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">e</font> = <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1;<br/>
<a NAME="E4:89"/><i><font color="Green" title="E68">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</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">P</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E5:89"/><i><font color="Green" title="E69">A5</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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">P</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="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </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="c2">P</font> )
 
<b>by </b><i><a class="txt" href="matrprob.html#E2:89"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="matrix_2.html#T12" title="MATRIX_2:th.12">MATRIX_2:12</a></i>;<br/>
<a NAME="E6:89"/><b>then </b><i><font color="Green" title="E70">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">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" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></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">p</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:89"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="matrprob.html#E4:89"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="matrixr1.html#T62" title="MATRIXR1:th.62">MATRIXR1:62</a></i>;<br/>
<a NAME="E7:89"/><i><font color="Green" title="E71">A7</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&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" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</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" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:89_1_1"/><i><font color="Green" title="E72">A8</font></i>: 
<font color="Maroon" title="c6">k</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:89_1_1"/><i><font color="Green" title="E73">A9</font></i>: 
<font color="Maroon" title="c6">k</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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:89_1_1"><i><font color="Green" title="E72">A8</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:89_1_1"/><b>then </b><i><font color="Green" title="E74">A10</font></i>: 
<font color="Maroon" title="c6">k</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="c2">P</font>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:89"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="matrprob.html#E6:89"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">k</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span>,<font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E4:89"><i><font color="Green" title="E68">A4</font></i></a>, <a class="txt" href="matrprob.html#E5:89"><i><font color="Green" title="E69">A5</font></i></a>, <a class="txt" href="matrprob.html#E2:89_1_1"><i><font color="Green" title="E73">A9</font></i></a>, <a class="ref" href="matrprob.html#T50" target="_self" title="MATRPROB:th.50">Th50</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:89_1_1"><i><font color="Green" title="E74">A10</font></i></a>, <a class="ref" href="matrix_2.html#T16" title="MATRIX_2:th.16">MATRIX_2:16</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:89_1_1"><i><font color="Green" title="E74">A10</font></i></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="matrprob.html#E3:89_1_1"><i><font color="Green" title="E74">A10</font></i></a>, <a class="ref" href="matrprob.html#D9" target="_self" title="MATRPROB:def.9">Def9</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" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1
 <b>by </b><i><a class="txt" href="matrprob.html#E2:89"><i><font color="Green" title="E66">A2</font></i></a>, <a class="txt" href="matrprob.html#E6:89"><i><font color="Green" title="E70">A6</font></i></a>, <a class="ref" href="matrprob.html#T1" target="_self" title="MATRPROB:th.1">Th1</a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green" title="E72">A11</font></i>:  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span> = 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="matrprob.html#E3:89"><i><font color="Green" title="E67">A3</font></i></a>, <a class="ref" href="matrprob.html#T32" target="_self" title="MATRPROB:th.32">Th32</a></i>
<br/>.= 
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c1">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" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="newton.html#K1" title="NEWTON:func.1">|-&gt;</a> 1</span>)</span> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K4" title="MATRIX_1:func.4">@</a> </span>)</span></span>)</span></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:89"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="matrprob.html#E2:89"><i><font color="Green" title="E66">A2</font></i></a>, <a class="txt" href="matrprob.html#E4:89"><i><font color="Green" title="E68">A4</font></i></a>, <a class="ref" href="matrprob.html#T49" target="_self" title="MATRPROB:th.49">Th49</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c1">p</font>
<b>by </b><i><a class="txt" href="matrprob.html#E7:89"><i><font color="Green" title="E71">A7</font></i></a>, <a class="ref" href="matrprob.html#T32" target="_self" title="MATRPROB:th.32">Th32</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrprob.html#D5" target="_self" title="MATRPROB:def.5">Def5</a></i>
;<br/>
<a NAME="E9:89"/>
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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<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" title="c6">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> ;<br/>

<b>assume </b><a NAME="E1:89_3"/><i><font color="Green" title="E73">A12</font></i>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span>
 ;<br/>

<a NAME="E2:89_3"/><i><font color="Green" title="E74">A13</font></i>: 
<font color="Maroon" title="c6">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> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E1:89_3"><i><font color="Green" title="E73">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:89_3"/><i><font color="Green" title="E75">A14</font></i>: 
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="c1">p</font> holds <br/><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="matrprob.html#D5" target="_self" title="MATRPROB:def.5">Def5</a></i>;<br/>
<a NAME="E4:89_3"/><i><font color="Green" title="E76">A15</font></i>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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 <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c2">P</font> holds <br/><font color="Maroon" title="c2">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="matrprob.html#D6" target="_self" title="MATRPROB:def.6">Def6</a></i>;<br/>
<a NAME="E5:89_3"/>
<font color="Maroon" title="c6">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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="matrprob.html#E3:89"><i><font color="Green" title="E67">A3</font></i></a>, <a class="txt" href="matrprob.html#E1:89_3"><i><font color="Green" title="E73">A12</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:89_3"/><b>then </b>
for <font color="Olive" title="b1">j</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">j</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> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="matrprob.html#E4:89_3"><i><font color="Green" title="E76">A15</font></i></a>, <a class="txt" href="matrprob.html#E22"><i><font color="Green" title="E22">Lm3</font></i></a></i>;<br/>
<a NAME="E7:89_3"/><b>then </b><i><font color="Green" title="E77">A16</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</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> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="matrprob.html#E3:89_3"><i><font color="Green" title="E75">A14</font></i></a>, <a class="ref" href="matrprob.html#T33" target="_self" title="MATRPROB:th.33">Th33</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> = 
<font color="Maroon" title="c1">p</font> <a href="matrprob.html#NK7" title="MATRPROB:NK.7">"*"</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="matrprob.html#E1:89"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="matrprob.html#E2:89_3"><i><font color="Green" title="E74">A13</font></i></a>, <a class="ref" href="matrprob.html#T40" target="_self" title="MATRPROB:th.40">Th40</a></i>
<br/>.= 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K14" title="RVSUM_1:func.14">mlt</a> <font color="Maroon" title="c1">p</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c6">i</font></span>)</span></span>)</span>

;<br/>
<b>hence </b><a NAME="E9:89_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="matrprob.html#E7:89_3"><i><font color="Green" title="E77">A16</font></i></a>, <a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:89"/>
( <font color="Maroon" title="c1">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</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>  &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">p</font> <a href="matrixr1.html#K12" title="MATRIXR1:func.12">*</a> <font color="Maroon" title="c2">P</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">P</font> )
 <b>by </b><i><a class="txt" href="matrprob.html#E2:89"><i><font color="Green" title="E66">A2</font></i></a>, <a class="txt" href="matrprob.html#E3:89"><i><font color="Green" title="E67">A3</font></i></a>, <a class="txt" href="matrprob.html#E8:89"><i><font color="Green" title="E72">A11</font></i></a>, <a class="ref" href="matrprob.html#D5" target="_self" title="MATRPROB:def.5">Def5</a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/>


</div>
