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

<b>let </b><font color="Maroon">V</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <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> <font color="Maroon">V</font>,<font color="Maroon">C</font></span>)</span>;<br/>





<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1">Function</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span> ;<br/>
<a NAME="E1:15"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">g</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">g</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">g</font>]
 
;<br/>
<a NAME="E2:15"/><i><font color="Green">E24</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">f</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S1">FRAENKEL:sch 1</a>();<br/></i>
<a NAME="E3:15"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E4:15"/><i><font color="Green">E44</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Olive">f</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<a NAME="E5:15"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font> iff  ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font>
 ;<br/><b>then 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="E3:15_1_1"/><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">x</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/>;<br/><a NAME="E4:15_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E5:15_1_1"/><b>then </b>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:15_1_1"/>
 ex <font color="Olive">Y</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">Y</font> &amp; <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span>  )
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">Y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><a NAME="E2:15_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span>  )
 ;<br/>

<b>consider </b><font color="Maroon">f1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f1</font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:15_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:15"/><b>then </b><i><font color="Green">E51</font></i>: 
 <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green">E52</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</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="E8:15"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span>  holds <br/><font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><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>assume </b><a NAME="E1:15_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) where <font color="Olive">f</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">f</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">f1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:15_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">f1</font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<b>thus </b><a NAME="E4:15_2"/>
<font color="Maroon">x</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, <a class="ref" href="finset_1.html#T29">FINSET_1:29</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:15"/>
 <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i>, , <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/>


</div>
