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

<span class="kw">let </span><font color="Maroon" title="c1">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>; <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#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</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="c1">M</font></span>)</span></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">p</font> =  <a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">q</font> =  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</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="c1">M</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:79"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</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="c1">M</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></span>;<br/>
<a NAME="E2:79"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></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> <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="c1">M</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<a NAME="E3:79"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></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> <span class="p1">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>;<br/>
<a NAME="E4:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></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">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#D8" target="_self" title="ENTROPY1:def.8">Def8</a></span>;<br/>
<a NAME="E5:79"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font></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">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="entropy1.html#T62" target="_self" title="ENTROPY1:th.62">Th62</a></span>;<br/>
<a NAME="E6:79"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:79"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="finseq_3.html#T29" title="FINSEQ_3:th.29">FINSEQ_3:29</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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="c4">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="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font></span>)</span> implies <span class="p1">(<span class="default"><a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:79_1"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
<font color="Maroon" title="c4">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="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</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"><span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="entropy1.html#T62" target="_self" title="ENTROPY1:th.62">Th62</a></span>
<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="matrprob.html#K3" title="MATRPROB:func.3">Sum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:79"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="txt" href="#E1:79_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a>, <a class="ref" href="matrprob.html#D1" title="MATRPROB:def.1">MATRPROB:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<span class="default"><a href="matrprob.html#NK4" title="MATRPROB:NK.4">LineSum</a> <span class="p3">(<span class="default"><a href="entropy1.html#K5" title="ENTROPY1:func.5">Infor_FinSeq_of</a> <font color="Maroon" title="c1">M</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rvsum_1.html#T17" title="RVSUM_1:th.17">RVSUM_1:17</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">hence </span><a NAME="E8:79"/>
 <a href="entropy1.html#K8" title="ENTROPY1:func.8">Entropy_of_Cond_Prob</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</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="c1">M</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:79"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="finseq_1.html#T13" title="FINSEQ_1:th.13">FINSEQ_1:13</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>
