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<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font> be  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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="entropy1.html#K6" title="ENTROPY1:func.6">Entropy</a> <font color="Maroon" title="c1">p</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><br/>

<b>set </b><font color="Maroon" title="c2">p1</font> =  <a href="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p1">(<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>;<br/>
<i><font color="Green" title="E58">A1</font></i>:  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">p</font> = 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <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>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<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>
<b>by </b><i><a class="ref" href="entropy1.html#T47" target="_self" title="ENTROPY1:th.47">Th47</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<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>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E2:69"/><i><font color="Green" title="E59">A2</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<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> holds <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="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">k</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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">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="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<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> implies <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="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</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><br/>

<b>assume </b><a NAME="E1:69_2"/><i><font color="Green" title="E60">A3</font></i>: 
<font color="Maroon" title="c3">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="rvsum_1.html#K6" title="RVSUM_1:func.6">-</a> <span class="p2">(<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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</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><br/>

<a NAME="E2:69_2"/>
<font color="Maroon" title="c3">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> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="txt" href="entropy1.html#E1:69"><i><font color="Green" title="E58">A1</font></i></a>, <a class="txt" href="entropy1.html#E1:69_2"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="valued_1.html#T8" title="VALUED_1:th.8">VALUED_1:8</a></i>;<br/>
<a NAME="E3:69_2"/><b>then </b>
<span class="p1">(<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> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="entropy1.html#T52" target="_self" title="ENTROPY1:th.52">Th52</a></i>;<br/>
<a NAME="E4:69_2"/><b>then </b>
 <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="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="real_1.html#K1" title="REAL_1:func.1">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T26" title="XREAL_1:th.26">XREAL_1:26</a></i>;<br/>
<b>hence </b><a NAME="E5:69_2"/>
<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="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">k</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="rvsum_1.html#T35" title="RVSUM_1:th.35">RVSUM_1:35</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>
<a NAME="E3:69"/>
 <a href="entropy1.html#K6" title="ENTROPY1:func.6">Entropy</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</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="entropy1.html#K4" title="ENTROPY1:func.4">Infor_FinSeq_of</a> <font color="Maroon" title="c1">p</font></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="E4:69"/>
 <a href="entropy1.html#K6" title="ENTROPY1:func.6">Entropy</a> <font color="Maroon" title="c1">p</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="txt" href="entropy1.html#E2:69"><i><font color="Green" title="E59">A2</font></i></a>, <a class="ref" href="rvsum_1.html#T114" title="RVSUM_1:th.114">RVSUM_1:114</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>
