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<b>let </b><font color="Maroon" title="c1">V</font>, <font color="Maroon" title="c2">C</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">A</font>, <font color="Maroon" title="c4">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1">(<span class="default"><a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font></span>)</span>;<br/>







<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1 <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> $2;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; $1 <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> $2 );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>,  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>] <b>means </b>( $2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; $2 <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> $1 );<br/>
<b>set </b><font color="Maroon" title="c5">X1</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">X2</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span> ;<br/>
<div><i><font color="Green" title="E5">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><a NAME="E1:12_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span>  iff  ex <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] ) )
 ;<br/><a NAME="E2:12_1"/><b>then </b>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span>  iff  ex <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> st <br/>( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">t</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b2">s</font> ) )
 ;<br/><b>hence </b><a NAME="E3:12_1"/>
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span>  iff <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">t</font>, <font color="Olive" title="b2">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : ( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> &amp; <font color="Olive" title="b2">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Olive" title="b1">t</font> <a href="partfun1.html#R1" title="PARTFUN1:pred.1">tolerates</a> <font color="Olive" title="b2">s</font> ) </span>}</span>  )
 ;<br/></div><b>end;</b></div>
<a NAME="E2:12"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] )
 
;<br/>
<a NAME="E3:12"/><i><font color="Green" title="E7">A3</font></i>: 
for <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>)
 
;<br/>
<a NAME="E4:12"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">t</font>,<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font>, <font color="Olive" title="b2">t</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">C</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>,<font color="Olive" title="b2">t</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S8" title="FRAENKEL:sch.8">FRAENKEL:sch 8</a>(<a class="txt" href="substlat.html#E2:12"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="substlat.html#E3:12"><i><font color="Green" title="E7">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon" title="c3">A</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">B</font> <a href="substlat.html#K4" title="SUBSTLAT:func.4">^</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="substlat.html#E1:12"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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