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

<b>let </b><font color="Maroon" title="c3">z1</font>, <font color="Maroon" title="c4">z2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font>;<br/>

<b>thus </b><a NAME="E1:48_1_2"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z2</font> )
 ;<br/>

<b>assume </b><a NAME="E2:48_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<a NAME="E3:48_1_2"/><b>then </b><i><font color="Green" title="E41">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>given </b><font color="Maroon" title="c5">p1</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><a NAME="E5:48_1_2"/><i><font color="Green" title="E42">A12</font></i>: 
( <font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">p1</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> &amp; <font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 &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="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">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="c2">M</font> holds <br/><font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/>

<b>given </b><font color="Maroon" title="c6">p2</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <font color="Maroon" title="c1">D</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><a NAME="E7:48_1_2"/><i><font color="Green" title="E43">A13</font></i>: 
( <font color="Maroon" title="c4">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">M</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p2</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> &amp; <font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> 1 &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="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">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="c2">M</font> holds <br/><font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">k</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span> ) )
 ;<br/>

<a NAME="E8:48_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="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Olive" title="b1">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="c2">M</font> holds <br/><font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>( $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; $1 <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="c2">M</font> implies <font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> $1 );<br/>
<a NAME="E1:48_1_2_1"/><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E2:48_1_2_1"/><i><font color="Green" title="E45">A15</font></i>: 
for <font color="Olive" title="b1">n</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">n</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="E1:48_1_2_1_1"/><i><font color="Green" title="E46">A16</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c7">n</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="c2">M</font> implies <font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c7">n</font> )
 ;<br/>

<b>assume </b><a NAME="E2:48_1_2_1_1"/><i><font color="Green" title="E47">A17</font></i>: 
( <font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1 <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="c2">M</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_1_2_1_1_1_1"/>
( <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:48_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:48_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="entropy1.html#E5:48_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="entropy1.html#E7:48_1_2"><i><font color="Green" title="E43">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_1_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:48_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c7">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:48_1_2_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E48">A18</font></i>: 
<font color="Maroon" title="c7">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:48_1_2_1_1_1_1_2"/><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c7">n</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="c2">M</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E2:48_1_2_1_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E1:48_1_2_1_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="entropy1.html#E2:48_1_2_1_1_1_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="entropy1.html#E5:48_1_2"><i><font color="Green" title="E42">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="entropy1.html#E2:48_1_2_1_1_1_1_2"><i><font color="Green" title="E48">A18</font></i></a>, <a class="txt" href="entropy1.html#E3:48_1_2_1_1_1_1_2"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="entropy1.html#E7:48_1_2"><i><font color="Green" title="E43">A13</font></i></a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:48_1_2_1_1"/>
<font color="Maroon" title="c5">p1</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p2</font> <a href="matrprob.html#K1" title="MATRPROB:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">n</font> <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> 1</span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E3:48_1_2_1"/>
for <font color="Olive" title="b1">n</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  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="entropy1.html#E1:48_1_2_1"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="entropy1.html#E2:48_1_2_1"><i><font color="Green" title="E45">A15</font></i></a>);</i><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:48_1_2"/>
<font color="Maroon" title="c3">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z2</font>
 <b>by </b><i><a class="txt" href="entropy1.html#E7:48_1_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="txt" href="entropy1.html#E5:48_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="entropy1.html#E3:48_1_2"><i><font color="Green" title="E41">A11</font></i></a></i>;<br/>


</div>
