<?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">X</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:102"/><i><font color="Green">E30</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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E2:102"/>
<font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:102"/><b>then </b>
<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T104">RELAT_1:104</a></i>;<br/>
<a NAME="E4:102"/><b>then </b>
 <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">Y</font></span>)</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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T25">RELAT_1:25</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">dfy</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">Y</font></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="finset_1.html#T13">FINSET_1:13</a></i>;<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">Y</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">Z</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st <font color="Olive">Z</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">Y</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="xboole_0.html#K2">\/</a> <font color="Olive">Y</font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Olive">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &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">Z</font> holds <br/> <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="xboole_0.html#K2">\/</a> <font color="Olive">Y</font></span>)</span>,<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#K8">^</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">Y</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> ;<br/>
<a NAME="E6:102"/><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="102_1"><b>proof </b></a><div class="add">

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



<b>assume </b><a NAME="E1:102_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">Z</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">Y</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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; 0 <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">Z</font> )
 ;<br/>

<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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> = 
<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> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<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#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">Y</font></span>)</span>
<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> <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#K4">\/</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">Y</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="E3:102_1"/><b>then </b>
(  <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="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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</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">Y</font></span>)</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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E4:102_1"/><b>then </b><i><font color="Green">E34</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> is <a href="finset_1.html#V1">finite</a> &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">Y</font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E5:102_1"/><i><font color="Green">E35</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">Y</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>,<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span> = 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<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>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#D4" target="_self">Def4</a>, <a class="ref" href="rfunct_3.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">X</font>
<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="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> <a href="numbers.html#K1">REAL</a> </span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</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="finseq_1.html#K8">^</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"><a href="relset_1.html#K4">dom</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i>, </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="finseq_1.html#K8">^</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">Y</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T10" target="_self">Th10</a></i>
;<br/>
<b>hence </b><a NAME="E7:102_1"/>
 <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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<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#K8">^</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">Y</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E7:102"/><i><font color="Green">E36</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="102_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="E1:102_2"/><i><font color="Green">E94</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/>

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



<b>assume </b><a NAME="E2:102_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">Z</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">Y</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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <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">Z</font> )
 ;<br/>

<i><font color="Green">E99</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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span> = 
<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> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">Y</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<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#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">Y</font></span>)</span>
<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> <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#K4">\/</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">Y</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of  <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">Y</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font> <b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </b><font color="Maroon">y1</font> = <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>;<br/>
<a NAME="E6:102_2"/>
<font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E7:102_2"/><b>then </b>
<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> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E8:102_2"/><b>then </b>
 <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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</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> <font color="Maroon">F</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E9:102_2"/><b>then </b>
 <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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</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> <span class="p2">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</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="E10:102_2"/><b>then </b><i><font color="Green">E100</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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span></span>)</span> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E11:102_2"/>
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E12:102_2"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E13:102_2"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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">Y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E14:102_2"/><i><font color="Green">E103</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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">Y</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">Y</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<a NAME="E15:102_2"/><i><font color="Green">E105</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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">Y</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">x</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:102_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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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">Y</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">x</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="102_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="E1:102_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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:102_2_2_1"/><b>then </b><i><font color="Green">E106</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:102_2_2_1"/><b>then </b><i><font color="Green">E107</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</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">x</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="E4:102_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">Y</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:102_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">Y</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">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <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="E2:102_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">Y</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">x</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:102_2_2"/><b>then </b><i><font color="Green">E108</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">Y</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">x</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="E4:102_2_2"/><b>then </b><i><font color="Green">E122</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">Y</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:102_2_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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/>
<b>hence </b><a NAME="E6:102_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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#T16" target="_self">Th16</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">dFy</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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="finset_1.html#T16">FINSET_1:16</a></i>;<br/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">dFy</font> = 
<span class="p1">(<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> <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">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
<b>by </b><i>, , , <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="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">n</font>

;<br/>
<a NAME="E18:102_2"/><b>then </b>
 <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="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span>,<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#K8">^</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <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>, , </i>;<br/>
<a NAME="E19:102_2"/><b>then </b>
<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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<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#K8">^</a> <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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#T14">RFINSEQ:14</a></i>;<br/>
<a NAME="E20:102_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<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#K8">^</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E21:102_2"/><i><font color="Green">E124</font></i>: 
<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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T12" target="_self">Th12</a>, <a class="ref" href="card_1.html#T47">CARD_1:47</a></i>;<br/>
<a NAME="E22:102_2"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">Y</font>,<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">Y</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E23:102_2"/><b>then </b>
<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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>, <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">Y</font> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T16" target="_self">Th16</a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E24:102_2"/><b>then </b><i><font color="Green">E128</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>,<span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</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"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">Y</font></span>)</span> <a href="finseq_1.html#K8">^</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="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfinseq.html#T14">RFINSEQ:14</a></i>;<br/>
<a NAME="E25:102_2"/><i><font color="Green">E129</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="finseq_1.html#K8">^</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_1.html#K8">^</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="rfinseq.html#R1">are_fiberwise_equipotent</a>  &amp; <span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">Y</font></span>)</span> <a href="finseq_1.html#K8">^</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"><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#K8">^</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">Y</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a>  )
 
<b>by </b><i><a class="ref" href="rfinseq.html#T15">RFINSEQ:15</a></i>;<br/>
<a NAME="E26:102_2"/><b>then </b>
<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#K8">^</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<font color="Maroon">Y</font></span>)</span> <a href="finseq_1.html#K8">^</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="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#T2">RFINSEQ:2</a></i>;<br/>
<a NAME="E27:102_2"/><b>then </b>
<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#K8">^</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">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>,<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#K8">^</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">Y</font></span>)</span> <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="E28:102_2"/><b>then </b><i><font color="Green">E130</font></i>: 
<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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<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#K8">^</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">Y</font></span>)</span> <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/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="102_2_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:102_2_4"/><i><font color="Green">E163</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:102_2_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E3:102_2_4"/>
contradiction
 <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>
<a NAME="E30:102_2"/><b>then </b>
<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">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T65">ZFMISC_1:65</a></i>;<br/>
<a NAME="E31:102_2"/><b>then </b><i><font color="Green">E164</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T42">XBOOLE_1:42</a></i>;<br/>
<a NAME="E32:102_2"/>
<font color="Maroon">x</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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="card_1.html#T47">CARD_1:47</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E33:102_2"/><b>then </b><i><font color="Green">E165</font></i>: 
<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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>,<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T19" target="_self">Th19</a>, <a class="ref" href="rfunct_3.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E34:102_2"/>
 <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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<font color="Maroon">F</font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i>, <a class="ref" href="rfunct_3.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E35:102_2"/><b>then </b>
<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="xboole_0.html#K2">\/</a> <span class="p3">(<span class="default"><font color="Maroon">Y</font> <a href="subset_1.html#K6">\</a> <span class="p4"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</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">x</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>, <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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 
<b>by </b><i><a class="ref" href="rfunct_3.html#T20" target="_self">Th20</a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>
<b>hence </b><a NAME="E36:102_2"/>
 <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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<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#K8">^</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">Y</font></span>)</span> <a href="rfinseq.html#R1">are_fiberwise_equipotent</a> 
 <b>by </b><i><a class="ref" href="rfunct_3.html#T18" target="_self">Th18</a>, <a class="ref" href="rfinseq.html#T2">RFINSEQ:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:102"/><i><font color="Green">E166</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>  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>assume </b><a NAME="E9:102"/><i><font color="Green">E167</font></i>: 
<font color="Maroon">X</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<a NAME="E10:102"/>
 <a href="card_1.html#K4">card</a> <font color="Maroon">dfy</font> <a href="hidden.html#R1">=</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">dfy</font>
 
;<br/>
<b>hence </b><a NAME="E11:102"/>
 <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="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span>,<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#K8">^</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">Y</font></span>)</span> <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>, </i>;<br/>


</div>
