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<b>let </b><font color="Maroon" title="c1">p</font> be  <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">M</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="E1:81"/><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">M</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c3">M1</font> = <span class="p1">(<span class="default"><a href="entropy1.html#K1" title="ENTROPY1:func.1">Vec2DiagMx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">M</font>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p2">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span>
<b>by </b><i><a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></i>
<br/>.= 
 <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="entropy1.html#E1:81"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="entropy1.html#D8" target="_self" title="ENTROPY1:def.8">Def8</a></i>
;<br/>
<a NAME="E3:81"/><b>then </b><i><font color="Green" title="E66">A2</font></i>: 
( <font color="Maroon" title="c1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  &amp;  <a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">M1</font> = <span class="p1">(<span class="default"><a href="entropy1.html#K1" title="ENTROPY1:func.1">Vec2DiagMx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">M</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>  <b>by </b><i><a class="txt" href="entropy1.html#E1:81"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="entropy1.html#T28" target="_self" title="ENTROPY1:th.28">Th28</a></i>;<br/>
<i><font color="Green" title="E67">A3</font></i>:  <a href="entropy1.html#K7" title="ENTROPY1:func.7">Entropy_of_Joint_Prob</a> <font color="Maroon" title="c4">M1</font> = 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="entropy1.html#K2" title="ENTROPY1:func.2">Mx2FinS</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c4">M1</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="entropy1.html#T59" target="_self" title="ENTROPY1:th.59">Th59</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><a href="matrprob.html#K5" title="MATRPROB:func.5">SumAll</a> <span class="p2">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c4">M1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="entropy1.html#T42" target="_self" title="ENTROPY1:th.42">Th42</a></i>
<br/>.= 
 <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<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="p4">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p5">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E1:81"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="entropy1.html#T64" target="_self" title="ENTROPY1:th.64">Th64</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<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="p3">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p4">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span></span>)</span>

;<br/>
<span class="p1">(<span class="default"><a href="entropy1.html#K6" title="ENTROPY1:func.6">Entropy</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<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="p3">(<span class="default"><a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<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="p3">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p4">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p5">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="entropy1.html#T63" target="_self" title="ENTROPY1:th.63">Th63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p3">(<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="p4">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p5">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E3:81"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="rvsum_1.html#T94" title="RVSUM_1:th.94">RVSUM_1:94</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p3">(<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="p4">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p5">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T118" title="RVSUM_1:th.118">RVSUM_1:118</a></i>
;<br/>
<b>hence </b><a NAME="E7:81"/>
 <a href="entropy1.html#K7" title="ENTROPY1:func.7">Entropy_of_Joint_Prob</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="entropy1.html#K1" title="ENTROPY1:func.1">Vec2DiagMx</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="matrixr1.html#K6" title="MATRIXR1:func.6">*</a> <font color="Maroon" title="c2">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="entropy1.html#K6" title="ENTROPY1:func.6">Entropy</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p2">(<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="p3">(<span class="default"><a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c2">M</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E5:81"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/>


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