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

<b>let </b><font color="Maroon" title="c1">D</font>, <font color="Maroon" title="c2">C</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">F</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c1">D</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/><b>let </b><font color="Maroon" title="c4">B</font> be   <a href="rearran1.html#NM1" title="REARRAN1:NM.1">RearrangmentGen</a> of <font color="Maroon" title="c2">C</font>;<br/>







<b>assume </b><a NAME="E1:35"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c3">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="c2">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="c1">D</font> )
 ;<br/>

<b>set </b><font color="Maroon" title="c5">fd</font> =  <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font>;<br/>
<b>set </b><font color="Maroon" title="c6">p</font> =  <a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</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="c3">F</font>,<font color="Maroon" title="c1">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="c4">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="c4">B</font></span>)</span>,<font color="Maroon" title="c2">C</font>;<br/>
<a NAME="E2:35"/><i><font color="Green" title="E32">A2</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="c3">F</font>,<font color="Maroon" title="c1">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="c4">B</font></span>)</span>,<font color="Maroon" title="c2">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="c4">B</font></span>)</span>,<font color="Maroon" title="c2">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="c4">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="c3">F</font>,<font color="Maroon" title="c1">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="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:35"><i><font color="Green" title="E31">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/>
<a NAME="E3:35"/><i><font color="Green" title="E33">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</font> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="rearran1.html#T21" target="_self" title="REARRAN1:th.21">Th21</a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></i>;<br/>
<a NAME="E4:35"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c3">F</font> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">|</a> <font color="Maroon" title="c1">D</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T97" title="RELAT_1:th.97">RELAT_1:97</a></i>;<br/>
<a NAME="E5:35"/><i><font color="Green" title="E35">A5</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span></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="E6:35"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">F</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">F'</font> = <font color="Maroon" title="c3">F</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="ref" href="finset_1.html#T29" title="FINSET_1:th.29">FINSET_1:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">dfd</font> =  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>, <font color="Maroon" title="c12">dF</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">F'</font> as  <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/>
<a NAME="E9:35"/>
 <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font>,<font color="Maroon" title="c3">F</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:35"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="rearran1.html#E4:35"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="rfunct_3.html#D14" title="RFUNCT_3:def.14">RFUNCT_3:def 14</a></i>;<br/>
<a NAME="E10:35"/><b>then </b>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">dfd</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="c12">dF</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c11">dfd</font> <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="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> &amp; <font color="Maroon" title="c1">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="rearran1.html#E5:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a>, <a class="ref" href="rfinseq.html#T10" title="RFINSEQ:th.10">RFINSEQ:10</a></i>;<br/>
<a NAME="E11:35"/><b>then </b>
 <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="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="rearran1.html#E3:35"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<a NAME="E12:35"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 
;<br/>
<a NAME="E13:35"/><b>then </b><i><font color="Green" title="E36">A6</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E14:35"/><b>then </b><i><font color="Green" title="E37">A7</font></i>: 
( 1 <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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>thus </b><a NAME="E15:35"/>
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">c</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font><b> such that </b><br/><a NAME="E3:35_1"/><i><font color="Green" title="E38">A8</font></i>: 
( <font color="Maroon" title="c14">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c14">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="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 );<br/>
<a NAME="E4:35_1"/><i><font color="Green" title="E39">A9</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c15">n</font> =  <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="c4">B</font></span>)</span>;<br/>

<a NAME="E1:35_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font>
 
<b>by </b><i><a class="ref" href="rearran1.html#T3" target="_self" title="REARRAN1:th.3">Th3</a></i>;<br/>
<b>hence </b><a NAME="E2:35_1_1"/>
( <font color="Maroon" title="c15">n</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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> &amp; <font color="Maroon" title="c14">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="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="rearran1.html#E13:35"><i><font color="Green" title="E36">A6</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c15">mi</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E6:35_1"/><i><font color="Green" title="E40">A10</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">mi</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c15">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E4:35_1"><i><font color="Green" title="E39">A9</font></i></a>);<br/></i>
<a NAME="E7:35_1"/><i><font color="Green" title="E41">A11</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font> &amp; <font color="Maroon" title="c15">mi</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="c4">B</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E6:35_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:35_1"/><b>then </b>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c15">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4" title="FINSEQ_2:th.4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">m1</font> = <font color="Maroon" title="c15">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>  <b>by </b><i><a class="ref" href="finseq_2.html#T5" title="FINSEQ_2:th.5">FINSEQ_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_1_2_1"/>
( <font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_1"><b>case </b></a><a NAME="E1:35_1_2_1_1"/>
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:35_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c3">F</font>,<font color="Maroon" title="c1">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#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="rearran1.html#E6:35_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E3:35_1_2_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E14:35"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="rearran1.html#E3:35_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_1_2_1_2"><b>case </b></a><a NAME="E1:35_1_2_1_2"/>
<font color="Maroon" title="c15">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:35_1_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">mi</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:35_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:35_1_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:35_1_2_1_2"/><b>then </b><i><font color="Green" title="E42">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">m1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E5:35_1_2_1_2"/><i><font color="Green" title="E43">A13</font></i>: 
<font color="Maroon" title="c16">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">mi</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E6:35_1_2_1_2"/><b>then </b><i><font color="Green" title="E44">A14</font></i>: 
<font color="Maroon" title="c16">m1</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="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:35_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:35_1_2_1_2"/>
<font color="Maroon" title="c16">m1</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="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E7:35_1"><i><font color="Green" title="E41">A11</font></i></a>, <a class="txt" href="rearran1.html#E5:35_1_2_1_2"><i><font color="Green" title="E43">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:35_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c16">m1</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="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E4:35_1_2_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:35_1_2_1_2"/><b>then </b>
not <font color="Maroon" title="c14">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="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E6:35_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E10:35_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c14">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="c4">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="c16">m1</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="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">m1</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E6:35_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E11:35_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c14">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="c3">F</font>,<font color="Maroon" title="c1">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="c16">m1</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#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="rearran1.html#E4:35_1_2_1_2"><i><font color="Green" title="E42">A12</font></i></a>, <a class="txt" href="rearran1.html#E6:35_1_2_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E12:35_1_2_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="rearran1.html#E5:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="rearran1.html#E3:35_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="rearran1.html#E6:35_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:35_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 ;<br/>


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

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E16:35"/><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( $1 <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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">x</font> );<br/>
<a NAME="E17:35"/><i><font color="Green" title="E39">A16</font></i>: 
 ex <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<b>by </b><i><a class="txt" href="rearran1.html#E16:35"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c14">mi</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:35"/><i><font color="Green" title="E40">A17</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c14">mi</font>] &amp; ( for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon" title="c14">mi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="rearran1.html#E17:35"><i><font color="Green" title="E39">A16</font></i></a>);<br/></i>
<a NAME="E20:35"/><i><font color="Green" title="E41">A18</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">mi</font> &amp; <font color="Maroon" title="c14">mi</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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="rearran1.html#E19:35"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E21:35"/><b>then </b>
 <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><font color="Maroon" title="c14">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T4" title="FINSEQ_2:th.4">FINSEQ_2:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c15">m1</font> = <font color="Maroon" title="c14">mi</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 as    <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>  <b>by </b><i><a class="ref" href="finseq_2.html#T5" title="FINSEQ_2:th.5">FINSEQ_2:5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:35_2_1"/>
( <font color="Maroon" title="c14">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c14">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_1"><b>case </b></a><a NAME="E1:35_2_1_1"/><i><font color="Green" title="E42">A19</font></i>: 
<font color="Maroon" title="c14">mi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:35_2_1_1"/><i><font color="Green" title="E43">A20</font></i>: 
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="rearran1.html#E5:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="rearran1.html#E14:35"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="rearran1.html#T7" target="_self" title="REARRAN1:th.7">Th7</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1;<br/><a NAME="E4:35_2_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="rearran1.html#E5:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="rearran1.html#E14:35"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green" title="E14">Lm5</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font> <b>by </b><i><a class="txt" href="rearran1.html#E2:35_2_1_1"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E6:35_2_1_1"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">y</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="c3">F</font>,<font color="Maroon" title="c1">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#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="rearran1.html#E2:35_2_1_1"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E7:35_2_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:35"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="rearran1.html#E19:35"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="rearran1.html#E1:35_2_1_1"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="35_2_1_2"><b>case </b></a><a NAME="E1:35_2_1_2"/>
<font color="Maroon" title="c14">mi</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:35_2_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">mi</font>
 <b>by </b><i><a class="txt" href="rearran1.html#E20:35"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:35_2_1_2"/><b>then </b>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">mi</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E4:35_2_1_2"/><b>then </b><i><font color="Green" title="E42">A21</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">m1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E5:35_2_1_2"/><i><font color="Green" title="E43">A22</font></i>: 
<font color="Maroon" title="c15">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">mi</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T46" title="XREAL_1:th.46">XREAL_1:46</a></i>;<br/><a NAME="E6:35_2_1_2"/><b>then </b><i><font color="Green" title="E44">A23</font></i>: 
<font color="Maroon" title="c15">m1</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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E20:35"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:35_2_1_2"/>
( <font color="Maroon" title="c15">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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> <span class="p1">(<span class="default"><a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> &amp; <font color="Maroon" title="c15">m1</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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E20:35"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="rearran1.html#E5:35_2_1_2"><i><font color="Green" title="E43">A22</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:35_2_1_2"/><b>then </b>
( <font color="Maroon" title="c15">m1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 &amp; <font color="Maroon" title="c15">m1</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="rfunct_3.html#K21" title="RFUNCT_3:func.21">FinS</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c1">D</font></span>)</span> )
 <b>by </b><i><a class="txt" href="rearran1.html#E4:35_2_1_2"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></i>;<br/><a NAME="E9:35_2_1_2"/><b>then </b><i><font color="Green" title="E45">A24</font></i>: 
<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="c4">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="c15">m1</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="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m1</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="rearran1.html#E4:35_2_1_2"><i><font color="Green" title="E42">A21</font></i></a>, <a class="ref" href="rearran1.html#T8" target="_self" title="REARRAN1:th.8">Th8</a></i>;<br/><b>consider </b><font color="Maroon" title="c16">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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="c4">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="c15">m1</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="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">m1</font></span>)</span>;<br/><a NAME="E11:35_2_1_2"/>
( <font color="Maroon" title="c16">y</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="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">m1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><a href="rearran1.html#K2" title="REARRAN1:func.2">Co_Gen</a> <font color="Maroon" title="c4">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">mi</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font> )
 <b>by </b><i><a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="rearran1.html#E5:35"><i><font color="Green" title="E35">A5</font></i></a>, <a class="txt" href="rearran1.html#E19:35"><i><font color="Green" title="E40">A17</font></i></a>, <a class="txt" href="rearran1.html#E9:35_2_1_2"><i><font color="Green" title="E45">A24</font></i></a>, <a class="txt" href="rearran1.html#E14"><i><font color="Green" title="E14">Lm5</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c17">y</font> = <font color="Maroon" title="c16">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">C</font> ;<br/><a NAME="E13:35_2_1_2"/>
<span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c17">y</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="c3">F</font>,<font color="Maroon" title="c1">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="c15">m1</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#E1:35"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="rearran1.html#E2:35"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="rearran1.html#E4:35_2_1_2"><i><font color="Green" title="E42">A21</font></i></a>, <a class="txt" href="rearran1.html#E6:35_2_1_2"><i><font color="Green" title="E44">A23</font></i></a>, <a class="txt" href="rearran1.html#E9:35_2_1_2"><i><font color="Green" title="E45">A24</font></i></a>, <a class="ref" href="rearran1.html#T22" target="_self" title="REARRAN1:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E14:35_2_1_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 <b>by </b><i><a class="txt" href="rearran1.html#E3:35"><i><font color="Green" title="E33">A3</font></i></a>, <a class="txt" href="rearran1.html#E19:35"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E24:35"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rearran1.html#K4" title="REARRAN1:func.4">Rlor</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c4">B</font></span>)</span>
 ;<br/>


</div>
