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

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><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    <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">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>assume </b><a NAME="E1:28"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="substlat.html#K4">^</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="heyting2.html#K3" target="_self">=&gt;&gt;</a> <font color="Maroon">K</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:28"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="heyting2.html#K3" target="_self">=&gt;&gt;</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E5:28"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="substlat.html#T15">SUBSTLAT:15</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">A</font>,<font color="Maroon">K</font><b> such that </b><br/><a NAME="E7:28"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</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">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span>  &amp;  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="txt" href="heyting2.html#E3:28"><i><font color="Green">E21</font></i></a>, </i>;<br/>
<a NAME="E8:28"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i>, , <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>
<a NAME="E9:28"/>
<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>
 
<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">d</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</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><a class="txt" href="heyting2.html#E4:28"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<b>thus </b><a NAME="E11:28"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="partfun1.html#T27">PARTFUN1:27</a></i>;<br/>

<a NAME="E12:28"/>
<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/>
<b>then reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</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="E14:28"/>
<font color="Maroon">d</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font></span>)</span> <a href="subset_1.html#K6">\</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</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">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E15:28"/><b>then </b>
<font color="Maroon">d</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E16:28"/>
<font color="Maroon">d</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T44">XBOOLE_1:44</a></i>;<br/>


</div>
