<?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">a</font> be  <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>;<br/>





<a NAME="E1:9"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">u</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E2:9"/>
for <font color="Olive">s</font>, <font color="Olive">t</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">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t</font> holds <br/><font color="Olive">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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>, <font color="Maroon">t</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>assume </b><a NAME="E1:9_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">s</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">t</font> )
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:9"/><b>then </b>
<span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">a</font></span><a href="setwiseo.html#K2">.}</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> 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> : ( ( for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">u</font> is <a href="finset_1.html#V1">finite</a> ) &amp; ( for <font color="Olive">s</font>, <font color="Olive">t</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">s</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">s</font> <a href="tarski.html#R1">c=</a> <font color="Olive">t</font> holds <br/><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="E4:9"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a</font></span><a href="tarski.html#K1">}</a></span> <a href="hidden.html#R2">in</a>  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="substlat.html#D1">SUBSTLAT:def 1</a></i>;<br/>


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