<?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">d</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font>;<br/>





<b>assume </b><a NAME="E1:93"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:93"/><b>then </b>
<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> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then </b><i><font color="Green">E30</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> = 
<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"><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_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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="funct_1.html#T68">FUNCT_1:68</a></i>
;<br/>
<b>then </b><i><font color="Green">E31</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>,<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="card_1.html#K4">card</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#T13" target="_self">Th13</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
;<br/>
<a NAME="E5:93"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<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>,<font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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> <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="E6:93"/><b>then </b><i><font color="Green">E32</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> <font color="Maroon">F</font>,<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>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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="rfinseq.html#T1">RFINSEQ:1</a></i>;<br/>
<a NAME="E7:93"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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"><a href="tarski.html#K1">{</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="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_3"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:93_3"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</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="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="93_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:93_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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>then consider </b><font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E3:93_3_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">e</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#K23" target="_self">|</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; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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="seq_1.html#K2">.</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<a NAME="E4:93_3_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:93_3_1"/><b>then </b>
<font color="Maroon">x</font> <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="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E6:93_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</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="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">x</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="E2:93_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</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="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:93_3"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">d</font> &amp; <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#K23" target="_self">|</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="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:93_3"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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="seq_1.html#K2">.</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/>
<b>hence </b><a NAME="E5:93_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="rfunct_3.html#K23" target="_self">|</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:93"/>
 <a href="rfunct_3.html#K22" target="_self">FinS</a> <font color="Maroon">F</font>,<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="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>
 <b>by </b><i>, <a class="ref" href="rfunct_3.html#D3" target="_self">Def3</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>


</div>
