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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>1</sub></font>] means not <font color="Maroon">a<sub>1</sub></font> <a href="normform.html#K1">\/</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> ;<br/>
<a NAME="E1:39"/><i><font color="Green">E38</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>5</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="heyting1.html#T10" target="_self">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<a NAME="E3:39"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="heyting1.html#E1:39"><i><font color="Green">E38</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:39"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K10" target="_self">SUB</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font>] means ( <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> ] means verum;<br/>
<a NAME="E5:39"/><i><font color="Green">E39</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; not <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  <a href="xboole_0.html#R1">misses</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></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">E40</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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">E41</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>7</sub></font>] )
 ;<br/>
<b>thus </b><a NAME="E4:39_1_1"/>
<font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E3:39_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  )
 ;<br/>
<b>thus </b><a NAME="E5:39_1"/>
<font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>2</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>2</sub></font>] </span>}</span>  &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E4:39_1"><i><font color="Green">E42</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:39"/><i><font color="Green">E42</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>6</sub></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">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:39_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>7</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>7</sub></font>] )
 ;<br/>
<b>thus </b><a NAME="E4:39_2_1"/>
<font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E3:39_2_1"><i><font color="Green">E43</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:39_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Maroon">c<sub>7</sub></font>) &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>7</sub></font>] )
 ;<br/>
<b>thus </b><a NAME="E5:39_2"/>
<font color="Maroon">c<sub>6</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 <b>by </b><i><a class="txt" href="heyting1.html#E4:39_2"><i><font color="Green">E43</font></i></a></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">c<sub>1</sub></font>,   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> <font color="Maroon">c<sub>1</sub></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> ] means ( <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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>2</sub></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> ] means ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  );<br/>
<a NAME="E9:39"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] iff <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">E44</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S4">FRAENKEL:sch 4</a>(<a class="txt" href="heyting1.html#E9:39"><i><font color="Green">E43</font></i></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> ] means <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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S20">FRAENKEL:sch 20</a>();<br/></i>
<b>hence </b><span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">c<sub>5</sub></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">c<sub>2</sub></font></span><a href="heyting1.html#K2" target="_self">}</a></span> = 
<span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="normform.html#K1">\/</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<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">b<sub>2</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) </span>}</span>  </span>}</span> 
<b>by </b><i><a class="txt" href="heyting1.html#E10:39"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i><a class="txt" href="heyting1.html#E5:39"><i><font color="Green">E39</font></i></a>, <a class="txt" href="heyting1.html#E6:39"><i><font color="Green">E40</font></i></a>, <a class="txt" href="heyting1.html#E7:39"><i><font color="Green">E41</font></i></a>, <a class="txt" href="heyting1.html#E8:39"><i><font color="Green">E42</font></i></a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/><br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E13:39"/>
<font color="Maroon">c<sub>6</sub></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">c<sub>3</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<a NAME="E14:39"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="ref" href="heyting1.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E15:39"/><b>then </b>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>6</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">b<sub>1</sub></font>) where B is    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font> : ( <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></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">c<sub>6</sub></font> <a href="normform.html#K1">\/</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div>
