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<b>let </b><font color="Maroon" title="c1">MR</font> be   <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_Probability</a> iff  <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a> )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a> implies <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_Probability</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_Probability</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a></span><br/><b>then reconsider </b><font color="Maroon" title="c2">MRR</font> = <font color="Maroon" title="c1">MR</font> as  <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_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/><a NAME="E3:54_1"/><i><font color="Green" title="E48">A1</font></i>: 
<font color="Maroon" title="c2">MRR</font> is  <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a> <a href="matrprob.html#V3" title="MATRPROB:attr.3">with_sum=1</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/><a NAME="E4:54_1"/><b>then </b>
 <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 <b>by </b><i><a class="ref" href="entropy1.html#T43" target="_self" title="ENTROPY1:th.43">Th43</a></i>;<br/><a NAME="E5:54_1"/><b>then </b><i><font color="Green" title="E49">A2</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> <span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</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="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/> <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font></span>)</span> = 
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">MR</font>
<b>by </b><i><a class="ref" href="entropy1.html#T42" target="_self" title="ENTROPY1:th.42">Th42</a></i>
<br/>.= 
1
<b>by </b><i><a class="txt" href="#E3:54_1"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="matrprob.html#D7" title="MATRPROB:def.7">MATRPROB:def 7</a></i>
;<br/><b>hence </b><a NAME="E7:54_1"/>
 <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>
 <b>by </b><i><a class="txt" href="#E5:54_1"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="matrprob.html#D5" title="MATRPROB:def.5">MATRPROB:def 5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:54"/>
 <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V1" title="MATRPROB:attr.1">ProbFinS</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_Probability</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">pp</font> =  <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> as  <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>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">pp</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>  ;<br/>
<a NAME="E5:54"/>
<font color="Maroon" title="c3">p</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 
;<br/>
<a NAME="E6:54"/><b>then </b><i><font color="Green" title="E48">A3</font></i>: 
<font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a>
 
<b>by </b><i><a class="ref" href="entropy1.html#T43" target="_self" title="ENTROPY1:th.43">Th43</a></i>;<br/>
 <a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <font color="Maroon" title="c1">MR</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c3">p</font>
<b>by </b><i><a class="ref" href="entropy1.html#T42" target="_self" title="ENTROPY1:th.42">Th42</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="matrprob.html#D5" title="MATRPROB:def.5">MATRPROB:def 5</a></i>
;<br/>
<a NAME="E8:54"/><b>then </b>
<font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V3" title="MATRPROB:attr.3">with_sum=1</a>
 
<b>by </b><i><a class="ref" href="matrprob.html#D7" title="MATRPROB:def.7">MATRPROB:def 7</a></i>;<br/>
<b>hence </b><a NAME="E9:54"/>
<font color="Maroon" title="c1">MR</font> is <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_Probability</a>
 <b>by </b><i><a class="txt" href="#E6:54"><i><font color="Green" title="E48">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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