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

<b>let </b><font color="Maroon" title="c1">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">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="c3">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c4">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c5">G</font> be  <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>assume </b><a NAME="E1:99"/><i><font color="Green" title="E64">A1</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c4">X</font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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>means </b>for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">G</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Olive" title="b1">X</font></span>)</span> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">G</font> holds <br/> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Olive" title="b1">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Olive" title="b1">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b2">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>;<br/>
<a NAME="E2:99"/><i><font color="Green" title="E65">A2</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c7">G</font> be  <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>assume </b><a NAME="E1:99_1"/><i><font color="Green" title="E66">A3</font></i>: 
<font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">X</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:99_1"/>
0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">G</font>
 ;<br/>

<a NAME="E3:99_1"/><b>then </b><i><font color="Green" title="E67">A4</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:99_1"><i><font color="Green" title="E66">A3</font></i></a>, <a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/>
<b>then </b> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">X</font> = 
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="ref" href="rfunct_3.html#T66" target="_self" title="RFUNCT_3:th.66">Th66</a></i>
<br/>.= 
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
<b>by </b><i><a class="ref" href="rfunct_3.html#T71" target="_self" title="RFUNCT_3:th.71">Th71</a></i>
;<br/>
<a NAME="E5:99_1"/><b>then </b><i><font color="Green" title="E68">A5</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<b>by </b><i><a class="ref" href="finseq_2.html#T87" title="FINSEQ_2:th.87">FINSEQ_2:87</a></i>;<br/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c6">X</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#D12" target="_self" title="RFUNCT_3:def.12">Def12</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c6">X</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b> <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c6">X</font> = 
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:99_1"><i><font color="Green" title="E67">A4</font></i></a>, <a class="ref" href="rfunct_3.html#T66" target="_self" title="RFUNCT_3:th.66">Th66</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c6">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c7">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E5:99_1"><i><font color="Green" title="E68">A5</font></i></a>, <a class="ref" href="rfunct_3.html#T71" target="_self" title="RFUNCT_3:th.71">Th71</a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:99"/><i><font color="Green" title="E66">A6</font></i>: 
for <font color="Olive" title="b1">n</font> being   <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>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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>assume </b><a NAME="E1:99_2"/><i><font color="Green" title="E67">A7</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c6">n</font>]
 ;<br/>

<b>let </b><font color="Maroon" title="c7">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c8">G</font> be  <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>assume </b><a NAME="E2:99_2"/><i><font color="Green" title="E68">A8</font></i>: 
<font color="Maroon" title="c8">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:99_2"/><i><font color="Green" title="E69">A9</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">G</font>
 ;<br/>

<i><font color="Green" title="E70">A10</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">X</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#D12" target="_self" title="RFUNCT_3:def.12">Def12</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E5:99_2"/><b>then </b><i><font color="Green" title="E71">A11</font></i>: 
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="rfunct_3.html#D14" target="_self" title="RFUNCT_3:def.14">Def14</a></i>;<br/>
<a NAME="E6:99_2"/><b>then </b><i><font color="Green" title="E72">A12</font></i>: 
 <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rvsum_1.html#K1" title="RVSUM_1:func.1">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1" title="RFINSEQ:th.1">RFINSEQ:1</a></i>;<br/>
<a NAME="E7:99_2"/><i><font color="Green" title="E73">A13</font></i>: 
0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">frx</font> =  <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font>;<br/>
<b>set </b><font color="Maroon" title="c10">fx</font> =  <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font>;<br/>
<a NAME="E8:99_2"/><i><font color="Green" title="E74">A14</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">G</font> &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#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">G</font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="ref" href="rfunct_3.html#T70" target="_self" title="RFUNCT_3:th.70">Th70</a></i>;<br/>
<a NAME="E9:99_2"/><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#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</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="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:99_2"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="rfunct_3.html#E7:99_2"><i><font color="Green" title="E73">A13</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E10:99_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span></span>)</span> <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#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">d</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font><b> such that </b><br/><a NAME="E12:99_2"/><i><font color="Green" title="E75">A15</font></i>: 
( <font color="Maroon" title="c11">d</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"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E6:99_2"><i><font color="Green" title="E72">A12</font></i></a>, <a class="ref" href="partfun1.html#T26" title="PARTFUN1:th.26">PARTFUN1:26</a></i>;<br/>
<a NAME="E13:99_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E14:99_2"/><b>then </b><i><font color="Green" title="E76">A16</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="rfunct_3.html#T75" target="_self" title="RFUNCT_3:th.75">Th75</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">Y</font> = <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<b>set </b><font color="Maroon" title="c13">dx</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c14">dy</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c15">fry</font> =  <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">fy</font> =  <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c17">n1r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font>;<br/>
<b>set </b><font color="Maroon" title="c18">nr</font> = <font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font>;<br/>
<a NAME="E15:99_2"/>
(  <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp;  <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E3:99_2"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="rfunct_3.html#E8:99_2"><i><font color="Green" title="E74">A14</font></i></a>, <a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green" title="E75">A15</font></i></a>, <a class="txt" href="rfunct_3.html#E14:99_2"><i><font color="Green" title="E76">A16</font></i></a>, <a class="ref" href="rfunct_3.html#T73" target="_self" title="RFUNCT_3:th.73">Th73</a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>;<br/>
<a NAME="E16:99_2"/><b>then </b><i><font color="Green" title="E77">A17</font></i>: 
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></i>;<br/>
<a NAME="E17:99_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</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#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></i>;<br/>
<a NAME="E18:99_2"/><b>then </b><i><font color="Green" title="E78">A18</font></i>: 
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:99_2"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="rfunct_3.html#E8:99_2"><i><font color="Green" title="E74">A14</font></i></a>, <a class="ref" href="rfinseq.html#T20" title="RFINSEQ:th.20">RFINSEQ:20</a></i>;<br/>
<a NAME="E19:99_2"/><i><font color="Green" title="E79">A19</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T33" title="RFINSEQ:th.33">RFINSEQ:33</a></i>;<br/>
<a NAME="E20:99_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="rfunct_3.html#T69" target="_self" title="RFUNCT_3:th.69">Th69</a></i>;<br/>
<a NAME="E21:99_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>, <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E5:99_2"><i><font color="Green" title="E71">A11</font></i></a>, <a class="ref" href="rfinseq.html#T2" title="RFINSEQ:th.2">RFINSEQ:2</a></i>;<br/>
<a NAME="E22:99_2"/><b>then </b>
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font> <a href="rfinseq.html#R1" title="RFINSEQ:pred.1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E18:99_2"><i><font color="Green" title="E78">A18</font></i></a>, <a class="ref" href="rfinseq.html#T14" title="RFINSEQ:th.14">RFINSEQ:14</a></i>;<br/>
<a NAME="E23:99_2"/><b>then </b><i><font color="Green" title="E80">A20</font></i>: 
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E19:99_2"><i><font color="Green" title="E79">A19</font></i></a>, <a class="ref" href="rfinseq.html#T36" title="RFINSEQ:th.36">RFINSEQ:36</a></i>;<br/>
<a NAME="E24:99_2"/><i><font color="Green" title="E81">A21</font></i>: 
( <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">X</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green" title="E70">A10</font></i></a>, <a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green" title="E75">A15</font></i></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E25:99_2"/><i><font color="Green" title="E82">A22</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:99_2_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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="99_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">y</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="E1:99_2_2_1"/>
<font color="Maroon" title="c19">y</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"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:99_2_2_1"/><b>then </b><i><font color="Green" title="E83">A23</font></i>: 
( <font color="Maroon" title="c19">y</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> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E24:99_2"><i><font color="Green" title="E81">A21</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:99_2_2_1"/><b>then </b><i><font color="Green" title="E84">A24</font></i>: 
( <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> &amp; not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:99_2_2_1"/>
<font color="Maroon" title="c19">y</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"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E24:99_2"><i><font color="Green" title="E81">A21</font></i></a>, <a class="txt" href="rfunct_3.html#E2:99_2_2_1"><i><font color="Green" title="E83">A23</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:99_2_2_1"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:99_2_2_1"><i><font color="Green" title="E84">A24</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c19">y</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="E2:99_2_2"/>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<a NAME="E3:99_2_2"/><b>then </b><i><font color="Green" title="E83">A25</font></i>: 
( <font color="Maroon" title="c19">y</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"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span> &amp; not <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:99_2_2"/><b>then </b><i><font color="Green" title="E84">A26</font></i>: 
( <font color="Maroon" title="c19">y</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> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E24:99_2"><i><font color="Green" title="E81">A21</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:99_2_2"/><b>then </b>
<font color="Maroon" title="c19">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E3:99_2_2"><i><font color="Green" title="E83">A25</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:99_2_2"/>
<font color="Maroon" title="c19">y</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"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E24:99_2"><i><font color="Green" title="E81">A21</font></i></a>, <a class="txt" href="rfunct_3.html#E4:99_2_2"><i><font color="Green" title="E84">A26</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c19">dx</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c7">X</font></span>)</span>, <font color="Maroon" title="c20">dy</font> =  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> 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>  <b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="ref" href="finset_1.html#T16" title="FINSET_1:th.16">FINSET_1:16</a></i>;<br/>
<i><font color="Green" title="E83">A27</font></i>:  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c20">dy</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c19">dx</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E24:99_2"><i><font color="Green" title="E81">A21</font></i></a>, <a class="txt" href="rfunct_3.html#E25:99_2"><i><font color="Green" title="E82">A22</font></i></a>, <a class="ref" href="card_2.html#T63" title="CARD_2:th.63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E3:99_2"><i><font color="Green" title="E69">A9</font></i></a>, <a class="ref" href="card_1.html#T79" title="CARD_1:th.79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon" title="c6">n</font>

;<br/>
<a NAME="E28:99_2"/><b>then </b><i><font color="Green" title="E84">A28</font></i>: 
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E1:99_2"><i><font color="Green" title="E67">A7</font></i></a></i>;<br/>
<a NAME="E29:99_2"/>
<font color="Maroon" title="c11">d</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"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green" title="E75">A15</font></i></a>, <a class="txt" href="rfunct_3.html#E24:99_2"><i><font color="Green" title="E81">A21</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D12" target="_self" title="RFUNCT_3:def.12">Def12</a></i>;<br/>
<a NAME="E31:99_2"/><b>then </b><i><font color="Green" title="E85">A29</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T50" title="RVSUM_1:th.50">RVSUM_1:50</a></i>;<br/>
<a NAME="E32:99_2"/><i><font color="Green" title="E86">A30</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E33:99_2"/><i><font color="Green" title="E87">A31</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &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#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> )
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green" title="E68">A8</font></i></a>, <a class="txt" href="rfunct_3.html#E3:99_2"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="rfunct_3.html#E27:99_2"><i><font color="Green" title="E83">A27</font></i></a>, <a class="ref" href="rfunct_3.html#T70" target="_self" title="RFUNCT_3:th.70">Th70</a></i>;<br/>
<a NAME="E34:99_2"/><b>then </b><i><font color="Green" title="E88">A32</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E32:99_2"><i><font color="Green" title="E86">A30</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E35:99_2"/><i><font color="Green" title="E89">A33</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E32:99_2"><i><font color="Green" title="E86">A30</font></i></a>, <a class="txt" href="rfunct_3.html#E33:99_2"><i><font color="Green" title="E87">A31</font></i></a>, <a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E36:99_2"/><i><font color="Green" title="E90">A34</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="binop_2.html#K34" title="BINOP_2:func.34">diffreal</a>  <a href="finseqop.html#K1" title="FINSEQOP:func.1">.:</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E37:99_2"/><b>then </b><i><font color="Green" title="E91">A35</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T86" title="FINSEQ_2:th.86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E38:99_2"/><b>then </b><i><font color="Green" title="E92">A36</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E35:99_2"><i><font color="Green" title="E89">A33</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E39:99_2"/><i><font color="Green" title="E93">A37</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#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</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#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</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="E40:99_2"/><i><font color="Green" title="E94">A38</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></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"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E41:99_2"/><i><font color="Green" title="E95">A39</font></i>: 
( <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E42:99_2"/><b>then </b><i><font color="Green" title="E96">A40</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"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E37:99_2"><i><font color="Green" title="E91">A35</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">m</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>assume </b><a NAME="E1:99_2_4"/><i><font color="Green" title="E97">A41</font></i>: 
<font color="Maroon" title="c21">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:99_2_4_1"/>
( <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c21">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_4_1_1"><b>suppose </b></a><a NAME="E1:99_2_4_1_1"/><i><font color="Green" title="E98">A42</font></i>: 
<font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:99_2_4_1_1"/><b>then </b><i><font color="Green" title="E99">A43</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E40:99_2"><i><font color="Green" title="E94">A38</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E3:99_2_4_1_1"/><i><font color="Green" title="E100">A44</font></i>: 
<font color="Maroon" title="c21">m</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"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E34:99_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="txt" href="rfunct_3.html#E1:99_2_4"><i><font color="Green" title="E97">A41</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">n</font></span>)</span>
<b>by </b><i><a class="txt" href="rfunct_3.html#E35:99_2"><i><font color="Green" title="E89">A33</font></i></a>, <a class="txt" href="rfunct_3.html#E38:99_2"><i><font color="Green" title="E92">A36</font></i></a>, <a class="txt" href="rfunct_3.html#E41:99_2"><i><font color="Green" title="E95">A39</font></i></a>, <a class="txt" href="rfunct_3.html#E1:99_2_4_1_1"><i><font color="Green" title="E98">A42</font></i></a>, <a class="ref" href="finseq_1.html#T37" title="FINSEQ_1:th.37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">r</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E36:99_2"><i><font color="Green" title="E90">A34</font></i></a>, <a class="txt" href="rfunct_3.html#E42:99_2"><i><font color="Green" title="E96">A40</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
;<br/><b>hence </b><a NAME="E5:99_2_4_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E14:99_2"><i><font color="Green" title="E76">A16</font></i></a>, <a class="txt" href="rfunct_3.html#E33:99_2"><i><font color="Green" title="E87">A31</font></i></a>, <a class="txt" href="rfunct_3.html#E1:99_2_4_1_1"><i><font color="Green" title="E98">A42</font></i></a>, <a class="txt" href="rfunct_3.html#E2:99_2_4_1_1"><i><font color="Green" title="E99">A43</font></i></a>, <a class="txt" href="rfunct_3.html#E3:99_2_4_1_1"><i><font color="Green" title="E100">A44</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="99_2_4_1_2"><b>suppose </b></a><a NAME="E1:99_2_4_1_2"/><i><font color="Green" title="E98">A45</font></i>: 
<font color="Maroon" title="c21">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:99_2_4_1_2"/><i><font color="Green" title="E99">A46</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">m</font> &amp; <font color="Maroon" title="c21">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:99_2_4"><i><font color="Green" title="E97">A41</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:99_2_4_1_2"/><b>then </b>
<font color="Maroon" title="c21">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:99_2_4_1_2"><i><font color="Green" title="E98">A45</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="E4:99_2_4_1_2"/><b>then </b><i><font color="Green" title="E100">A47</font></i>: 
<font color="Maroon" title="c21">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</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="E5:99_2_4_1_2"/><b>then </b><i><font color="Green" title="E101">A48</font></i>: 
<font color="Maroon" title="c21">m</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="c6">n</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2_4_1_2"><i><font color="Green" title="E99">A46</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:99_2_4_1_2"/><b>then </b><i><font color="Green" title="E102">A49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E35:99_2"><i><font color="Green" title="E89">A33</font></i></a>, <a class="txt" href="rfunct_3.html#E40:99_2"><i><font color="Green" title="E94">A38</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="E7:99_2_4_1_2"/><i><font color="Green" title="E103">A50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E5:99_2_4_1_2"><i><font color="Green" title="E101">A48</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E8:99_2_4_1_2"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> &amp; <font color="Maroon" title="c6">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E2:99_2_4_1_2"><i><font color="Green" title="E99">A46</font></i></a>, <a class="txt" href="rfunct_3.html#E4:99_2_4_1_2"><i><font color="Green" title="E100">A47</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E9:99_2_4_1_2"/><b>then </b>
<font color="Maroon" title="c6">n</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> <span class="p1">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:99_2_4_1_2"/><b>then </b><i><font color="Green" title="E104">A51</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">n</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> &amp; <font color="Maroon" title="c21">m</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#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E33:99_2"><i><font color="Green" title="E87">A31</font></i></a>, <a class="txt" href="rfunct_3.html#E39:99_2"><i><font color="Green" title="E93">A37</font></i></a>, <a class="txt" href="rfunct_3.html#E1:99_2_4"><i><font color="Green" title="E97">A41</font></i></a>, <a class="txt" href="rfunct_3.html#E5:99_2_4_1_2"><i><font color="Green" title="E101">A48</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="rfinseq.html#T19" title="RFINSEQ:th.19">RFINSEQ:19</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c22">s</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> ;<br/><a NAME="E12:99_2_4_1_2"/><i><font color="Green" title="E105">A52</font></i>: 
<font color="Maroon" title="c21">m</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"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E34:99_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="txt" href="rfunct_3.html#E1:99_2_4"><i><font color="Green" title="E97">A41</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="E13:99_2_4_1_2"/>
<font color="Maroon" title="c21">m</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"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E35:99_2"><i><font color="Green" title="E89">A33</font></i></a>, <a class="txt" href="rfunct_3.html#E5:99_2_4_1_2"><i><font color="Green" title="E101">A48</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p5"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c11">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c11">d</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c3">r</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font> = 
<font color="Maroon" title="c22">s</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c3">r</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E16:99_2"><i><font color="Green" title="E77">A17</font></i></a>, <a class="txt" href="rfunct_3.html#E6:99_2_4_1_2"><i><font color="Green" title="E102">A49</font></i></a>, <a class="txt" href="rfunct_3.html#E7:99_2_4_1_2"><i><font color="Green" title="E103">A50</font></i></a>, <a class="txt" href="rfunct_3.html#E10:99_2_4_1_2"><i><font color="Green" title="E104">A51</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c21">m</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E10:99_2_4_1_2"><i><font color="Green" title="E104">A51</font></i></a>, <a class="txt" href="rfunct_3.html#E12:99_2_4_1_2"><i><font color="Green" title="E105">A52</font></i></a>, <a class="ref" href="seq_1.html#D4" title="SEQ_1:def.4">SEQ_1:def 4</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E44:99_2"/>
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c7">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c7">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c8">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E3:99_2"><i><font color="Green" title="E69">A9</font></i></a>, <a class="txt" href="rfunct_3.html#E18:99_2"><i><font color="Green" title="E78">A18</font></i></a>, <a class="txt" href="rfunct_3.html#E23:99_2"><i><font color="Green" title="E80">A20</font></i></a>, <a class="txt" href="rfunct_3.html#E28:99_2"><i><font color="Green" title="E84">A28</font></i></a>, <a class="txt" href="rfunct_3.html#E31:99_2"><i><font color="Green" title="E85">A29</font></i></a>, <a class="txt" href="rfunct_3.html#E34:99_2"><i><font color="Green" title="E88">A32</font></i></a>, <a class="txt" href="rfunct_3.html#E38:99_2"><i><font color="Green" title="E92">A36</font></i></a>, <a class="ref" href="finseq_2.html#T10" title="FINSEQ_2:th.10">FINSEQ_2:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:99"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>2</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="rfunct_3.html#E2:99"><i><font color="Green" title="E65">A2</font></i></a>, <a class="txt" href="rfunct_3.html#E3:99"><i><font color="Green" title="E66">A6</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:99"/>
 <a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="rfunct_3.html#K21" title="RFUNCT_3:func.21">-</a> <font color="Maroon" title="c3">r</font></span>)</span>,<font color="Maroon" title="c4">X</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" title="RFUNCT_3:func.22">FinS</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c4">X</font></span>)</span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c5">G</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E1:99"><i><font color="Green" title="E64">A1</font></i></a></i>;<br/>


</div>
