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

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">D</font>, <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>









<b>assume </b><a NAME="E1:99"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">X</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">G</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st <font color="Olive">G</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Olive">X</font></span>)</span> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Olive">G</font> holds <br/> <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Olive">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Olive">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Olive">G</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span>;<br/>
<a NAME="E2:99"/><i><font color="Green">E31</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">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:99_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:99_1"/>
0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">G</font>
 ;<br/>

<a NAME="E3:99_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/>
<b>then </b> <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font> = 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="ref" href="rfunct_3.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> 
<b>by </b><i/>
;<br/>
<a NAME="E5:99_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">G</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T49">RVSUM_1:49</a></i>;<br/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>hence </b> <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font> = 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T10" target="_self">Th10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">G</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E3:99"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</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">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:99_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">G</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E2:99_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E3:99_2"/><i><font color="Green">E95</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">G</font>
 ;<br/>

<i><font color="Green">E99</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<a NAME="E5:99_2"/><b>then </b><i><font color="Green">E100</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, <a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E6:99_2"/><b>then </b><i><font color="Green">E101</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/>
<a NAME="E7:99_2"/><i><font color="Green">E102</font></i>: 
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>set </b><font color="Maroon">frx</font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font>;<br/>
<b>set </b><font color="Maroon">fx</font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font>;<br/>
<a NAME="E8:99_2"/><i><font color="Green">E103</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">G</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">G</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, , <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E9:99_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">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" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">d</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E12:99_2"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E13:99_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E14:99_2"/><b>then </b><i><font color="Green">E106</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, , , </i>;<br/>
<b>set </b><font color="Maroon">Y</font> = <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<b>set </b><font color="Maroon">dx</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">dy</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">fry</font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">fy</font> =  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span>;<br/>
<b>set </b><font color="Maroon">n1r</font> = <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font>;<br/>
<b>set </b><font color="Maroon">nr</font> = <font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font>;<br/>
<a NAME="E15:99_2"/>
(  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp;  <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, , , , , , , <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<a NAME="E16:99_2"/><b>then </b><i><font color="Green">E107</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/>
<a NAME="E17:99_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E18:99_2"/><b>then </b><i><font color="Green">E108</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i>, , <a class="ref" href="rfinseq.html#T20">RFINSEQ:20</a></i>;<br/>
<a NAME="E19:99_2"/><i><font color="Green">E122</font></i>: 
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> is <a href="rfinseq.html#V1">non-increasing</a>
 
<b>by </b><i><a class="ref" href="rfinseq.html#T33">RFINSEQ:33</a></i>;<br/>
<a NAME="E20:99_2"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, , , <a class="ref" href="rfunct_3.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E21:99_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>, <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E22:99_2"/><b>then </b>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rfinseq.html#T14">RFINSEQ:14</a></i>;<br/>
<a NAME="E23:99_2"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="rfinseq.html#T36">RFINSEQ:36</a></i>;<br/>
<a NAME="E24:99_2"/><i><font color="Green">E124</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font> &amp;  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="subset_1.html#K8">/\</a> <font color="Maroon">X</font> )
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E25:99_2"/><i><font color="Green">E128</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</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">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">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">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:99_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:99_2_2_1"/><b>then </b><i><font color="Green">E129</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:99_2_2_1"/><b>then </b><i><font color="Green">E130</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:99_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="rfunct_3.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:99_2_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


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

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

<b>assume </b><a NAME="E2:99_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:99_2_2"/><b>then </b><i><font color="Green">E163</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span> &amp; not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E4:99_2_2"/><b>then </b><i><font color="Green">E164</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:99_2_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E6:99_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">dx</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">X</font></span>)</span>, <font color="Maroon">dy</font> =  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> as  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, <a class="ref" href="finset_1.html#T16">FINSET_1:16</a></i>;<br/>
<i><font color="Green">E165</font></i>:  <a href="card_1.html#K4">card</a> <font color="Maroon">dy</font> = 
<span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">dx</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i><a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="rfunct_3.html#T20" target="_self">Th20</a>, <a class="ref" href="card_2.html#T63">CARD_2:63</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, , <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">n</font>

;<br/>
<a NAME="E28:99_2"/><b>then </b><i><font color="Green">E166</font></i>: 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E29:99_2"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E30:99_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="txt" href="rfunct_3.html#E10"><i><font color="Green">Lemma45</font></i></a></i>;<br/>
<a NAME="E31:99_2"/><b>then </b><i><font color="Green">E167</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="rvsum_1.html#T50">RVSUM_1:50</a></i>;<br/>
<a NAME="E32:99_2"/><i><font color="Green">E168</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>;<br/>
<a NAME="E33:99_2"/><i><font color="Green">E169</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, , <a class="ref" href="rfunct_3.html#T21" target="_self">Th21</a>, <a class="ref" href="rfunct_3.html#T13" target="_self">Th13</a></i>;<br/>
<a NAME="E34:99_2"/><i><font color="Green">E170</font></i>: 
( <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K34">diffreal</a>  <a href="finseqop.html#K1">.:</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K34">diffreal</a>  <a href="finseqop.html#K1">.:</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="rvsum_1.html#D6">RVSUM_1:def 6</a></i>;<br/>
<a NAME="E35:99_2"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T23" target="_self">Th23</a>, <a class="ref" href="rfunct_3.html#T24" target="_self">Th24</a>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E36:99_2"/><i><font color="Green">E172</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T23" target="_self">Th23</a>, <a class="ref" href="rfunct_3.html#T24" target="_self">Th24</a>, <a class="ref" href="rfunct_3.html#T26" target="_self">Th26</a>, <a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E37:99_2"/><i><font color="Green">E173</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="seq_1.html#K2">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K34">diffreal</a>  <a href="finseqop.html#K1">.:</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a>, <a class="ref" href="rvsum_1.html#D6">RVSUM_1:def 6</a></i>;<br/>
<a NAME="E38:99_2"/><b>then </b><i><font color="Green">E174</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_2.html#T86">FINSEQ_2:86</a></i>;<br/>
<a NAME="E39:99_2"/><b>then </b><i><font color="Green">E175</font></i>: 
 <a href="finseq_1.html#K3">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" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E40:99_2"/><i><font color="Green">E176</font></i>: 
 <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E41:99_2"/><i><font color="Green">E177</font></i>: 
(  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> &amp;  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E42:99_2"/><i><font color="Green">E210</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E43:99_2"/><b>then </b><i><font color="Green">E211</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">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">m</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:99_2_4"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</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">E213</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:99_2_4_1_1"/><b>then </b><i><font color="Green">E214</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/><a NAME="E3:99_2_4_1_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="finseq_1.html#D3">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" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green">E94</font></i></a>, , <a class="txt" href="rfunct_3.html#E5:99_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="finseq_1.html#T37">FINSEQ_1:37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i>, , <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</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" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</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" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T24" target="_self">Th24</a>, <a class="txt" href="rfunct_3.html#E5:99_2"><i><font color="Green">E100</font></i></a>, <a class="txt" href="rfunct_3.html#E6:99_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="rfunct_3.html#E7:99_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</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">E216</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:99_2_4_1_2"/><i><font color="Green">E217</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E3:99_2_4_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="rfunct_3.html#E5:99_2"><i><font color="Green">E100</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E4:99_2_4_1_2"/><b>then </b><i><font color="Green">E218</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:99_2_4_1_2"/><b>then </b><i><font color="Green">E227</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <font color="Maroon">n</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E6:99_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E6:99_2_4_1_2"/><b>then </b><i><font color="Green">E228</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" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E7:99_2_4_1_2"/><i><font color="Green">E229</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="rfunct_3.html#E8:99_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a></i>;<br/><a NAME="E8:99_2_4_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="rfunct_3.html#E6:99_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="rfunct_3.html#E7:99_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E9:99_2_4_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><a NAME="E10:99_2_4_1_2"/><b>then </b><i><font color="Green">E230</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">n</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="rfunct_3.html#T24" target="_self">Th24</a>, <a class="txt" href="rfunct_3.html#E3:99_2"><i><font color="Green">E95</font></i></a>, <a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green">E99</font></i></a>, <a class="txt" href="rfunct_3.html#E8:99_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_2.html#T70">FINSEQ_2:70</a>, <a class="ref" href="rfinseq.html#T19">RFINSEQ:19</a></i>;<br/><b>reconsider </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E12:99_2_4_1_2"/><i><font color="Green">E231</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="rfunct_3.html#E4:99_2"><i><font color="Green">E99</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E13:99_2_4_1_2"/>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="setwop_2.html#K3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p3">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="rfunct_3.html#E8:99_2"><i><font color="Green">E103</font></i></a>, <a class="ref" href="finseq_1.html#D3">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" target="_self">FinS</a> <font color="Maroon">F</font>,<span class="p4">(<span class="default"><font color="Maroon">X</font> <a href="subset_1.html#K6">\</a> <span class="p5"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">d</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rvsum_1.html#K7">-</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">r</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font> = 
<font color="Maroon">s</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, , <a class="txt" href="rfunct_3.html#E12:99_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="rfunct_3.html#E14:99_2"><i><font color="Green">E106</font></i></a>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">m</font>
<b>by </b><i><a class="txt" href="rfunct_3.html#E14:99_2"><i><font color="Green">E106</font></i></a>, <a class="txt" href="rfunct_3.html#E16:99_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="rvsum_1.html#T47">RVSUM_1:47</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E45:99_2"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">G</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T17" target="_self">Th17</a>, <a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, , <a class="ref" href="rfunct_3.html#T22" target="_self">Th22</a>, , <a class="txt" href="rfunct_3.html#E2:99_2"><i><font color="Green">E94</font></i></a>, <a class="ref" href="finseq_2.html#T10">FINSEQ_2:10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:99"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E5:99"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K21" target="_self">-</a> <font color="Maroon">r</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font></span>)</span> <a href="rvsum_1.html#K7">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">G</font></span>)</span> <a href="finseqop.html#K4">|-&gt;</a> <font color="Maroon">r</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
