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

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="heyting3.html#K1" target="_self">SubstPoset</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/><b>let </b><font color="Maroon">a'</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4">PFuncs</a> <a href="numbers.html#K5">NAT</a> ,<span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">k</font></span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being  <a href="finset_1.html#V1">finite</a> <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:57"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="heyting3.html#E2"><i><font color="Green">Lemma39</font></i></a>, </i>;<br/>
<a NAME="E5:57"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="heyting3.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/>
<b>then consider </b><font color="Maroon">k1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:57"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>thus </b><a NAME="E8:57"/>
 <a href="heyting2.html#K1">Involved</a> <font color="Maroon">a'</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="heyting3.html#E3"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="heyting3.html#E5"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="heyting3.html#T1" target="_self">Th1</a>, <a class="ref" href="heyting2.html#T10">HEYTING2:10</a>, <a class="ref" href="heyting2.html#T12">HEYTING2:12</a>, <a class="ref" href="heyting2.html#D1">HEYTING2:def 1</a></i>;<br/>


</div>
