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<b>let </b><font color="Maroon">M</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/>

<a NAME="E1:82"/><i><font color="Green">E43</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">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="E2:82"/><i><font color="Green">E48</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/>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"><font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Olive">i</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Olive">i</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_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="E1:82_1"/><i><font color="Green">E49</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/>

<a NAME="E2:82_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">j</font> being  <a href="nat_1.html#NM1">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"><font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span> holds <br/><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> <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>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="E3:82_1"/><i><font color="Green">E51</font></i>: 
<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"><font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>

<a NAME="E4:82_1"/>
<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> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <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#T6" target="_self">Th6</a>, , </i>;<br/>
<b>hence </b><a NAME="E5:82_1"/>
<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> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="txt" href="matrprob.html#E5:82"><i><font color="Green">E60</font></i></a></i>;<br/>


</div><b>end;</b></div>
<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="E3:82"/><i><font color="Green">E58</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">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>
 ;<br/>

<a NAME="E4:82"/><i><font color="Green">E59</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">M</font></span>)</span> &amp; <font color="Maroon">j</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">M</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, </i>;<br/>
<a NAME="E5:82"/><b>then </b><i><font color="Green">E60</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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:82"/><b>then </b>
<font color="Maroon">j</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">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="matrprob.html#T6" target="_self">Th6</a>, <a class="ref" href="gobrd13.html#T4">GOBRD13:4</a></i>;<br/>
<a NAME="E7:82"/><b>then </b><i><font color="Green">E61</font></i>: 
<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"><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="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E9:82"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="matrprob.html#K1" target="_self">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">j</font> )
 <b>by </b><i><a class="ref" href="matrprob.html#T5" target="_self">Th5</a>, <a class="ref" href="matrix_1.html#D6">MATRIX_1:def 6</a></i>;<br/>
<b>thus </b><a NAME="E10:82"/>
<font color="Maroon">M</font> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i>, , , <a class="ref" href="matrprob.html#T4" target="_self">Th4</a></i>;<br/>


</div>
