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

<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : 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="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Olive">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font>] <b>means </b>for <font color="Olive">h</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="Olive">h</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Maroon">a<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">h</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<a NAME="E1:19_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">v</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">v</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">v</font>]
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E2:19_1_1"/><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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<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="E3:19_1_1"/>
 <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/>;<br/>
<a NAME="E4:19_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
 <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:19_1_1"/>
<span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : verum </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:19_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : verum </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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:19_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>
<b>thus </b><a NAME="E4:19_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E6:19_1_1"/><b>then </b>
<span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : verum </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="E7:19_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : 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="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Olive">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</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="E8:19_1_1"/>
<span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : 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="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Olive">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</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="E1:19_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : 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="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Olive">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:19_1_1_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">f1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; ( 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="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Maroon">f1</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font> ) )
 ;<br/>
<a NAME="E4:19_1_1_2"/>
 <a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:19_1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</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="partfun1.html#T125">PARTFUN1:125</a></i>;<br/>
<a NAME="E6:19_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:19_1_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="partfun1.html#K4">PFuncs</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:19_1_1"/>
<span class="p1">{<span class="default"> <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> <span class="p1">(<span class="default"><a href="heyting2.html#K1" target="_self">Involved</a> <font color="Maroon">A</font></span>)</span>,<font color="Maroon">C</font> : 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="Olive">g</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> holds <br/> not <font color="Olive">f</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">g</font> </span>}</span>  is    <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>
 <b>by </b><i>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>


</div>
