<?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">a</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">A</font></span>)</span>;<br/>





<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>] <b>means </b>not <font color="Maroon">a<sub>1</sub></font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>;<br/>
<b>set </b><font color="Maroon">M</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] ) </span>}</span> ;<br/>
<a NAME="E1:39"/><i><font color="Green">E21</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">u</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S17">FRAENKEL:sch 17</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">v</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] ) </span>}</span>  as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12">NormForm</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">v</font>
;<br/>

<a NAME="E3:39"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">u</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K10" target="_self">SUB</a> <font color="Maroon">u</font>
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">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">\/</a> <font color="Maroon">a</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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">u</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>] <b>means </b>( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<a NAME="E5:39"/><i><font color="Green">E23</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>] &amp; not <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> ) </span>}</span>  <a href="xboole_0.html#R1">misses</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S18">FRAENKEL:sch 18</a>();<br/></i>
<a NAME="E6:39"/><i><font color="Green">E52</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></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>4</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s</font>] ) </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S14">FRAENKEL:sch 14</a>();<br/></i>
<a NAME="E7:39"/><i><font color="Green">E56</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></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>4</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s1</font> where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] </span>}</span>  </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">t</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s1</font> where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] </span>}</span>  </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">t</font>] ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">t</font>) &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">t</font>] )
 ;<br/>
<b>thus </b><a NAME="E4:39_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s1</font> where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E9"><i><font color="Green">Lemma42</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t</font>) where <font color="Olive">t</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">t</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">t</font>) &amp; <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] </span>}</span>  )
 ;<br/>
<b>thus </b><a NAME="E5:39_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">t1</font>) where <font color="Olive">t1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s1</font> where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">t1</font>] ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E9"><i><font color="Green">Lemma42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:39"/><i><font color="Green">E61</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s1</font>) where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s1</font>] ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s2</font>) where <font color="Olive">s2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s2</font>] </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:39_2"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s1</font>) where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s1</font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s2</font>) where <font color="Olive">s2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s2</font>] </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s1</font>) where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s1</font>] ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E3:39_2_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">s1</font>) &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">s1</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">s1</font>] )
 ;<br/>
<b>thus </b><a NAME="E4:39_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s2</font>) where <font color="Olive">s2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s2</font>] </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s1</font>) where <font color="Olive">s1</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s1</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">s1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font><b> such that </b><br/><a NAME="E4:39_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">s1</font>) &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">s1</font>] )
 ;<br/>
<b>thus </b><a NAME="E5:39_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">s2</font>) where <font color="Olive">s2</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s2</font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">s2</font>] ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>deffunc </b><font color="Maroon">H<sub>5</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>,   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">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">\/</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>5</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> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] ) </span>}</span>  &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="heyting1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">a</font></span><a href="heyting1.html#K2" target="_self">}</a></span> );<br/>
<b>defpred </b><font color="Maroon">S<sub>6</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#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] ) </span>}</span>  );<br/>
<a NAME="E9:39"/><i><font color="Green">E65</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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] iff <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:39"/><i><font color="Green">E66</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</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">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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>5</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>5</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">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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">s'</font>,<font color="Olive">t'</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S4">FRAENKEL:sch 4</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>7</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> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] ) </span>}</span> ;<br/>
<a NAME="E11:39"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</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">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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">S<sub>7</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>5</sub></font>(<font color="Olive">s'</font>,<font color="Maroon">a</font>) where <font color="Olive">s'</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">s'</font>,<font color="Maroon">a</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S20">FRAENKEL:sch 20</a>();<br/></i>
<b>hence </b><a NAME="E12:39"/>
<span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span> <a href="normform.html#K10">^</a> <span class="p1"><a href="heyting1.html#K2" target="_self">{</a><span class="default"><font color="Maroon">a</font></span><a href="heyting1.html#K2" target="_self">}</a></span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="heyting1.html#T3" target="_self">Th3</a>, <a class="txt" href="heyting1.html#E6"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="heyting1.html#E8"><i><font color="Green">Lemma41</font></i></a>, <a class="txt" href="heyting1.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>

<b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>;<br/>

<b>assume </b><a NAME="E13:39"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K11" target="_self">diff</a> <font color="Maroon">u</font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">v</font>
 ;<br/>

<a NAME="E14:39"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> <a href="subset_1.html#K6">\</a> <font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<a NAME="E15:39"/><b>then </b>
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></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">DISJOINT_PAIRS</a> <font color="Maroon">A</font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">s</font>) <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] ) </span>}</span>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E16:39"/>
<font color="Maroon">b</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 ;<br/>


</div>
