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







<b>assume </b><a NAME="E1:30"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="heyting1.html#K7" target="_self">=&gt;&gt;</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:30"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K8">/\</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="normform.html#K6">FinPairUnion</a> <font color="Maroon">B</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="heyting1.html#K5" target="_self">pair_diff</a> <font color="Maroon">A</font></span>)</span> <a href="funcop_1.html#K6">.:</a> <font color="Olive">f</font>,<span class="p4">(<span class="default"><a href="funct_3.html#K6">incl</a> <span class="p5">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>,<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="Olive">f</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> </span>}</span> 
 
;<br/>
<a NAME="E3:30"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="normform.html#K6">FinPairUnion</a> <font color="Maroon">B</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="heyting1.html#K5" target="_self">pair_diff</a> <font color="Maroon">A</font></span>)</span> <a href="funcop_1.html#K6">.:</a> <font color="Olive">f</font>,<span class="p4">(<span class="default"><a href="funct_3.html#K6">incl</a> <span class="p5">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span></span>)</span> where <font color="Olive">f</font> is    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>,<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="Olive">f</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:30"/><b>then </b>
 ex <font color="Olive">f</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>,<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> st <br/>( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="normform.html#K6">FinPairUnion</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" target="_self">pair_diff</a> <font color="Maroon">A</font></span>)</span> <a href="funcop_1.html#K6">.:</a> <font color="Olive">f</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Olive">f</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
;<br/>
<b>hence </b><a NAME="E5:30"/>
 ex <font color="Olive">f</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span>,<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> st <br/>( <font color="Olive">f</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="normform.html#K6">FinPairUnion</a> <font color="Maroon">B</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="heyting1.html#K5" target="_self">pair_diff</a> <font color="Maroon">A</font></span>)</span> <a href="funcop_1.html#K6">.:</a> <font color="Olive">f</font>,<span class="p2">(<span class="default"><a href="funct_3.html#K6">incl</a> <span class="p3">(<span class="default"><a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> )
 ;<br/>


</div>
