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

<b>let </b><font color="Maroon" title="c1">p</font> be    <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>assume </b><a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">p</font> is <a href="partfun3.html#V4" title="PARTFUN3:attr.4">nonnegative</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">k1</font> be    <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> ;<br/>

<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c2">k1</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> &amp; <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c1">p</font> )
 ;<br/>

<a NAME="E3:18"/><i><font color="Green" title="E18">A3</font></i>: 
( <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="entropy1.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">j1</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E5:18"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c2">k1</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">j1</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="c1">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">j1</font> = <font color="Maroon" title="c3">j1</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E7:18"/>
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>   holds <br/>(  not <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> <font color="Maroon" title="c1">p</font> or  not <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k1</font> or  not <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
 ex <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 <br/>( <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> <font color="Maroon" title="c1">p</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k1</font> &amp; <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">k2</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> <b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green" title="E20">A5</font></i>: 
( <font color="Maroon" title="c5">k2</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> &amp; <font color="Maroon" title="c5">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">k1</font> &amp; <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 ;<br/>
<a NAME="E4:18_1"/><i><font color="Green" title="E21">A6</font></i>: 
( <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="entropy1.html#E3:18_1"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">j2</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:18_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c5">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">j2</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="c1">p</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">j2</font> = <font color="Maroon" title="c6">j2</font> as    <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>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:18_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="entropy1.html#E3:18_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="entropy1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_1"/>
( <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">k2</font> or <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">k2</font> )
 </span><b>by </b><i><a class="txt" href="entropy1.html#E3:18_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_1_1"/>
<font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">k2</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_1"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
not <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</font> being    <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> <b> such that </b><br/><a NAME="E4:18_1_1_1"/><i><font color="Green" title="E25">A10</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">p2</font> )
 <b>by </b><i><a class="txt" href="entropy1.html#E6:18_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></i>;<br/><a NAME="E5:18_1_1_1"/>
<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a>, <a class="txt" href="entropy1.html#E4:18_1"><i><font color="Green" title="E21">A6</font></i></a></i>;<br/><a NAME="E6:18_1_1_1"/><b>then </b><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c5">k2</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="c8">p1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:18_1_1_1"/>
( <font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0 &amp; ( for <font color="Olive" title="b1">k</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">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="c8">p1</font> holds <br/><font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_1_1_1_1"/>
<font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="txt" href="entropy1.html#E6:18_1_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="entropy1.html#E8:18_1"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c10">k</font> be    <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> ;<br/>

<b>assume </b><a NAME="E2:18_1_1_1_1"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c10">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="c8">p1</font>
 ;<br/>

<a NAME="E3:18_1_1_1_1"/><i><font color="Green" title="E28">A13</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E4:18_1_1_1_1"/>
<font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="txt" href="entropy1.html#E2:18_1_1_1_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:18_1_1_1_1"/>
<font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E3:18_1_1_1_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="entropy1.html#E2:18_1_1_1_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="entropy1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_1_1_1"/><b>then </b><i><font color="Green" title="E27">A14</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c8">p1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E6:18_1_1_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="rvsum_1.html#T115" title="RVSUM_1:th.115">RVSUM_1:115</a></i>;<br/><a NAME="E9:18_1_1_1"/>
not <font color="Maroon" title="c2">k1</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="c8">p1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="entropy1.html#E2:18_1_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">kk</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> <b> such that </b><br/><a NAME="E11:18_1_1_1"/><i><font color="Green" title="E28">A15</font></i>: 
( <font color="Maroon" title="c10">kk</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="c9">p2</font> &amp; <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c10">kk</font> )
 <b>by </b><i><a class="txt" href="entropy1.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a>, <a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a></i>;<br/><a NAME="E12:18_1_1_1"/><i><font color="Green" title="E29">A16</font></i>: 
<font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="entropy1.html#E11:18_1_1_1"><i><font color="Green" title="E28">A15</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E13:18_1_1_1"/>
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> <font color="Maroon" title="c9">p2</font> holds <br/><font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:18_1_1_1_2"/><i><font color="Green" title="E30">A17</font></i>: 
<font color="Maroon" title="c11">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="c9">p2</font>
 ;<br/>

<a NAME="E2:18_1_1_1_2"/>
( <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j2</font> )
 
<b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="entropy1.html#E1:18_1_1_1_2"><i><font color="Green" title="E30">A17</font></i></a></i>;<br/>
<a NAME="E3:18_1_1_1_2"/><b>then </b>
( <font color="Maroon" title="c5">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0 &amp; <font color="Maroon" title="c5">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#E6:18_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:18_1_1_1_2"/><b>then </b><i><font color="Green" title="E31">A18</font></i>: 
<font color="Maroon" title="c5">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">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="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:18_1_1_1_2"/>
<font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k2</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c11">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="entropy1.html#E1:18_1_1_1_2"><i><font color="Green" title="E30">A17</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1_1_2"/>
<font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_1_2"><i><font color="Green" title="E31">A18</font></i></a>, <a class="txt" href="entropy1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:18_1_1_1"/><b>then </b><i><font color="Green" title="E30">A19</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">p2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E12:18_1_1_1"><i><font color="Green" title="E29">A16</font></i></a>, <a class="ref" href="matrprob.html#T5" title="MATRPROB:th.5">MATRPROB:5</a>, <a class="txt" href="entropy1.html#E11:18_1_1_1"><i><font color="Green" title="E28">A15</font></i></a></i>;<br/><a NAME="E15:18_1_1_1"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">p2</font></span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/><a NAME="E16:18_1_1_1"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E8:18_1_1_1"><i><font color="Green" title="E27">A14</font></i></a>, <a class="txt" href="entropy1.html#E14:18_1_1_1"><i><font color="Green" title="E30">A19</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E17:18_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_1_2"/>
<font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c5">k2</font>
 ;<br/><div class="add"><a NAME="E2:18_1_1_2"/><b>then </b><i><font color="Green" title="E24">A20</font></i>: 
not <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</font> being    <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> <b> such that </b><br/><a NAME="E4:18_1_1_2"/><i><font color="Green" title="E25">A21</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j1</font> &amp; <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c9">p2</font> )
 <b>by </b><i><a class="txt" href="entropy1.html#E5:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="finseq_2.html#T26" title="FINSEQ_2:th.26">FINSEQ_2:26</a></i>;<br/><a NAME="E5:18_1_1_2"/>
<font color="Maroon" title="c2">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a>, <a class="txt" href="entropy1.html#E3:18"><i><font color="Green" title="E18">A3</font></i></a></i>;<br/><a NAME="E6:18_1_1_2"/><b>then </b><i><font color="Green" title="E26">A22</font></i>: 
<font color="Maroon" title="c2">k1</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="c8">p1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E7:18_1_1_2"/>
( <font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font> &amp; ( 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> <font color="Maroon" title="c8">p1</font> holds <br/><font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0 ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:18_1_1_2_1"/>
<font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="txt" href="entropy1.html#E6:18_1_1_2"><i><font color="Green" title="E26">A22</font></i></a>, <a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a></i>;<br/>

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:18_1_1_2_1"/><i><font color="Green" title="E27">A23</font></i>: 
<font color="Maroon" title="c10">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="c8">p1</font>
 ;<br/>

<a NAME="E3:18_1_1_2_1"/><i><font color="Green" title="E28">A24</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">p1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></i>;<br/>
<a NAME="E4:18_1_1_2_1"/>
<font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a>, <a class="txt" href="entropy1.html#E2:18_1_1_2_1"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:18_1_1_2_1"/>
<font color="Maroon" title="c8">p1</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E2:18_1_1_2_1"><i><font color="Green" title="E27">A23</font></i></a>, <a class="txt" href="entropy1.html#E3:18_1_1_2_1"><i><font color="Green" title="E28">A24</font></i></a>, <a class="txt" href="entropy1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:18_1_1_2"/><b>then </b><i><font color="Green" title="E27">A25</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c8">p1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E6:18_1_1_2"><i><font color="Green" title="E26">A22</font></i></a>, <a class="ref" href="matrprob.html#T5" title="MATRPROB:th.5">MATRPROB:5</a></i>;<br/><a NAME="E9:18_1_1_2"/>
not <font color="Maroon" title="c5">k2</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="c8">p1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="entropy1.html#E2:18_1_1_2"><i><font color="Green" title="E24">A20</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">kk</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> <b> such that </b><br/><a NAME="E11:18_1_1_2"/><i><font color="Green" title="E28">A26</font></i>: 
( <font color="Maroon" title="c10">kk</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="c9">p2</font> &amp; <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c10">kk</font> )
 <b>by </b><i><a class="txt" href="entropy1.html#E3:18_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a>, <a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a></i>;<br/><a NAME="E12:18_1_1_2"/><i><font color="Green" title="E29">A27</font></i>: 
<font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c10">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="entropy1.html#E11:18_1_1_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E13:18_1_1_2"/>
for <font color="Olive" title="b1">k</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">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="c9">p2</font> holds <br/><font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">k</font> be    <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> ;<br/>

<b>assume </b><a NAME="E1:18_1_1_2_2"/><i><font color="Green" title="E30">A28</font></i>: 
<font color="Maroon" title="c11">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="c9">p2</font>
 ;<br/>

<a NAME="E2:18_1_1_2_2"/>
( <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j1</font> )
 
<b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="txt" href="entropy1.html#E1:18_1_1_2_2"><i><font color="Green" title="E30">A28</font></i></a></i>;<br/>
<a NAME="E3:18_1_1_2_2"/><b>then </b>
( <font color="Maroon" title="c2">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 0 &amp; <font color="Maroon" title="c2">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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#E5:18"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E4:18_1_1_2_2"/><b>then </b><i><font color="Green" title="E31">A29</font></i>: 
<font color="Maroon" title="c2">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c11">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="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E5:18_1_1_2_2"/>
<font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k1</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <font color="Maroon" title="c11">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="txt" href="entropy1.html#E1:18_1_1_2_2"><i><font color="Green" title="E30">A28</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_1_2_2"/>
<font color="Maroon" title="c9">p2</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_2_2"><i><font color="Green" title="E31">A29</font></i></a>, <a class="txt" href="entropy1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="entropy1.html#D1" target="_self" title="ENTROPY1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:18_1_1_2"/><b>then </b><i><font color="Green" title="E30">A30</font></i>: 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">p2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E11:18_1_1_2"><i><font color="Green" title="E28">A26</font></i></a>, <a class="txt" href="entropy1.html#E12:18_1_1_2"><i><font color="Green" title="E29">A27</font></i></a>, <a class="txt" href="entropy1.html#E8:18_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="rvsum_1.html#T115" title="RVSUM_1:th.115">RVSUM_1:115</a></i>;<br/><a NAME="E15:18_1_1_2"/>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c8">p1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <span class="p1">(<span class="default"><a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c9">p2</font></span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E4:18_1_1_2"><i><font color="Green" title="E25">A21</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>;<br/><a NAME="E16:18_1_1_2"/><b>then </b>
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c2">k1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> 0
 <b>by </b><i><a class="txt" href="entropy1.html#E8:18_1_1_2"><i><font color="Green" title="E27">A25</font></i></a>, <a class="txt" href="entropy1.html#E14:18_1_1_2"><i><font color="Green" title="E30">A30</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E17:18_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="entropy1.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E8:18"/>
<font color="Maroon" title="c1">p</font> <a href="entropy1.html#R1" title="ENTROPY1:pred.1">has_onlyone_value_in</a> <font color="Maroon" title="c2">k1</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="entropy1.html#D2" target="_self" title="ENTROPY1:def.2">Def2</a></i>;<br/>


</div>
