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







<b>assume </b><a NAME="E1:18"/><i><font color="Green">E19</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; ( for <font color="Olive">s</font> being <a href="finset_1.html#V1">finite</a>  <a href="hidden.html#M1">set</a>   st <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">s</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> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <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>;<br/>

<b>thus </b><a NAME="E1:18_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> implies <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s</font> as  <a href="finset_1.html#V1">finite</a>  <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>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E3:18_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 ;<br/>


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

<b>thus </b><a NAME="E2:18_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> implies ( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font> ) )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:18"/>
<font color="Maroon">B</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/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as    <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>by </b><i/>;<br/>
<a NAME="E5:18"/>
<font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</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">t</font> is <a href="finset_1.html#V1">finite</a> &amp; ( for <font color="Olive">s</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> holds <br/> ( ( <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t</font> ) iff <font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> ) ) ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:18"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K3" target="_self">mi</a> <font color="Maroon">B</font>
 ;<br/>


</div>
