<?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="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">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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>,   <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>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1">Function</a>,  <a href="funct_1.html#NM1">Function</a>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">a<sub>2</sub></font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="funct_1.html#NM1">Function</a>,  <a href="funct_1.html#NM1">Function</a>] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="partfun1.html#R1">tolerates</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<b>set </b><font color="Maroon">X1</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>,<font color="Olive">t</font>) where <font color="Olive">s</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="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">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">X2</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>) where <font color="Olive">s</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="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">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:12_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>) where <font color="Olive">s</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="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">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  iff  ex <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 <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] ) )
 ;<br/><a NAME="E2:12_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>) where <font color="Olive">s</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="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">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  iff  ex <font color="Olive">t</font>, <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> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">s</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">s</font> ) )
 ;<br/><b>hence </b><a NAME="E3:12_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>) where <font color="Olive">s</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="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">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  iff <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">s</font></span>)</span> 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">s</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> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">s</font> ) </span>}</span>  )
 ;<br/></div><b>end;</b></div>
<a NAME="E2:12"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>) where <font color="Olive">s</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="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">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</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">s</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> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">t</font> <a href="partfun1.html#R1">tolerates</a> <font color="Olive">s</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green">E23</font></i>: 
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> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] )
 
;<br/>
<a NAME="E4:12"/><i><font color="Green">E28</font></i>: 
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> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>,<font color="Olive">t</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>)
 
;<br/>
<a NAME="E5:12"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">s</font>,<font color="Olive">t</font>) where <font color="Olive">s</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="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">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">t</font>,<font color="Olive">s</font>) where <font color="Olive">s</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="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">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S8">FRAENKEL:sch 8</a>(, );<br/></i>
<b>hence </b><a NAME="E6:12"/>
<font color="Maroon">A</font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="substlat.html#K4" target="_self">^</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div>
