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

<b>let </b><font color="Maroon" title="c1">C</font>, <font color="Maroon" title="c2">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c5">B</font> be   <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c1">C</font>;<br/>









<b>set </b><font color="Maroon" title="c6">fd</font> =  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c7">mf</font> =  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">b</font> =  <a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font>;<br/>
<b>set </b><font color="Maroon" title="c9">h</font> =  <a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c1">C</font>;<br/>
<b>assume </b><a NAME="E1:34"/><i><font color="Green" title="E30">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> is <a href="partfun1.html#V1" title="PARTFUN1:attr.1">total</a> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c2">D</font> )
 ;<br/>

<a NAME="E2:34"/><i><font color="Green" title="E31">A2</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</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"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p4">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T35" title="RFUNCT_3:th.35">RFUNCT_3:35</a>, <a class="ref" href="rfunct_3.html#T36" title="RFUNCT_3:th.36">RFUNCT_3:36</a></i>;<br/>
<a NAME="E3:34"/><i><font color="Green" title="E32">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></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="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c1">C</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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:34"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="rearran1.html#T12" target="_self" title="REARRAN1:th.12">Th12</a>, <a class="ref" href="rfinseq.html#D3" title="RFINSEQ:def.3">RFINSEQ:def 3</a>, <a class="ref" href="rfunct_3.html#D6" title="RFUNCT_3:def.6">RFUNCT_3:def 6</a></i>;<br/>
<b>thus </b><a NAME="E4:34"/>
( <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 implies <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font> = 
 <a href="rvsum_1.html#K18" title="RVSUM_1:func.18">Sum</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E1:34"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="rearran1.html#E2:34"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="rearran1.html#T14" target="_self" title="REARRAN1:th.14">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="rearran1.html#E3:34"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="rearran1.html#T4" target="_self" title="REARRAN1:th.4">Th4</a>, <a class="ref" href="rfinseq.html#T29" title="RFINSEQ:th.29">RFINSEQ:29</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c10">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>set </b><font color="Maroon" title="c11">mfn</font> =  <a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span>;<br/>
<b>assume </b><a NAME="E5:34"/><i><font color="Green" title="E33">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">n</font> &amp; <font color="Maroon" title="c10">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> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span> )
 ;<br/>

<a NAME="E6:34"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="rfunct_3.html#K17" title="RFUNCT_3:func.17">(#)</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K16" title="RFUNCT_3:func.16">CHI</a> <span class="p3">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c5">B</font></span>)</span>,<font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="rfunct_3.html#K18" title="RFUNCT_3:func.18">#</a> <font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:34"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="rearran1.html#T14" target="_self" title="REARRAN1:th.14">Th14</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c5">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">c</font> = 
<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"><font color="Maroon" title="c10">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </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="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E2:34"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="rvsum_1.html#T105" title="RVSUM_1:th.105">RVSUM_1:105</a></i>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="rfinseq.html#K3" title="RFINSEQ:func.3">MIM</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <font color="Maroon" title="c10">n</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T111" title="RVSUM_1:th.111">RVSUM_1:111</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="rearran1.html#E3:34"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="rearran1.html#E5:34"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="rfinseq.html#T30" title="RFINSEQ:th.30">RFINSEQ:30</a></i>
;<br/><br/>


</div>
