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<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">K</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="substlat.html#K1">SubstitutionSet</a> <font color="Maroon">V</font>,<font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>







<b>assume </b><a NAME="E1:30"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K</font>
 ;<br/>

<a NAME="E2:30"/>
( <font color="Maroon">K</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> &amp; <font color="Maroon">K</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E3:30"/><b>then </b>
( <font color="Maroon">X</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> &amp; <font color="Maroon">X</font> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:30"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">X</font> as    <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/>
<a NAME="E5:30"/><i><font color="Green">E24</font></i>: 
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> <font color="Maroon">X</font> holds <br/><font color="Olive">x</font> is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:30"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E6:30"/>
for <font color="Olive">a</font>, <font color="Olive">b</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">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">a</font> <a href="tarski.html#R1">c=</a> <font color="Olive">b</font> holds <br/><font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font>
 
<b>by </b><i><a class="txt" href="heyting2.html#E1:30"><i><font color="Green">E21</font></i></a>, <a class="ref" href="substlat.html#T5">SUBSTLAT:5</a></i>;<br/>
<a NAME="E7:30"/><b>then </b>
<font color="Maroon">X</font> <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="E8:30"/>
<font color="Maroon">X</font> <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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