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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">then </span><span class="kw">reconsider </span><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"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
<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"/><span class="kw">then </span>
 <a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <font color="Maroon" title="c1">MR</font> is  <a href="entropy1.html#NV1" title="ENTROPY1:NV.1">nonnegative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#T43" target="_self" title="ENTROPY1:th.43">Th43</a></span>;<br/><a NAME="E5:54_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</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> <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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></span>;<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#T42" target="_self" title="ENTROPY1:th.42">Th42</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="matrprob.html#D7" title="MATRPROB:def.7">MATRPROB:def 7</a></span>
;<br/><span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_1"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="matrprob.html#D5" title="MATRPROB:def.5">MATRPROB:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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 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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#T42" target="_self" title="ENTROPY1:th.42">Th42</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#D5" title="MATRPROB:def.5">MATRPROB:def 5</a></span>
;<br/>
<a NAME="E6:54"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A3</font></span>: 
<font color="Maroon" title="c1">MR</font> is  <a href="matrprob.html#V3" title="MATRPROB:attr.3">with_sum=1</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#D7" title="MATRPROB:def.7">MATRPROB:def 7</a></span>;<br/>
<a NAME="E7:54"/>
<font color="Maroon" title="c3">p</font> is  <a href="entropy1.html#NV1" title="ENTROPY1:NV.1">nonnegative</a> 
 
;<br/>
<a NAME="E8:54"/><span class="kw">then </span>
<font color="Maroon" title="c1">MR</font> is  <a href="matrprob.html#V2" title="MATRPROB:attr.2">m-nonnegative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#T43" target="_self" title="ENTROPY1:th.43">Th43</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:54"/>
<font color="Maroon" title="c1">MR</font> is  <a href="matrprob.html#V4" title="MATRPROB:attr.4">Joint_Probability</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:54"><span class="lab"><font color="Green" title="E48">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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