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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<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="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>;<br/><b>let </b><font color="Maroon">C</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="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>;<br/>





<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">A</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> = <font color="Maroon">a<sub>1</sub></font> <a href="normform.html#K1" target="_self">\/</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span> ;<br/>
<a NAME="E1:56"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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="E2:56"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">s</font>, <font color="Olive">t</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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="E3:56"/><i><font color="Green">E44</font></i>: 
<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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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>(<a class="ref" href="normform.html#T39" target="_self">Th39</a>, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_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="E1:56_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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Olive">s</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 ;<br/><a NAME="E2:56_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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Olive">s</font> &amp; <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) )
 ;<br/><b>hence </b><a NAME="E3:56_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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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="normform.html#K1" target="_self">\/</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">s</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  )
 ;<br/></div><b>end;</b></div>
<a NAME="E5:56"/><b>then </b>
<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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</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"> <span class="p2">(<span class="default"><font color="Olive">t</font> <a href="normform.html#K1" target="_self">\/</a> <font color="Olive">s</font></span>)</span> where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">s</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E6:56"/><b>then </b>
( <font color="Maroon">B</font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  &amp; <font color="Maroon">C</font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</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#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font>, <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7" target="_self">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  )
 
;<br/>
<b>hence </b><a NAME="E7:56"/>
<font color="Maroon">B</font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font> <a href="normform.html#K10" target="_self">^</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="normform.html#E3:56"><i><font color="Green">E44</font></i></a></i>;<br/>


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