<?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>let </b><font color="Maroon">v</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>assume </b><b>that </b><br/><a NAME="E1:33"/>
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> holds <br/><font color="Olive">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>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> holds <br/> ex <font color="Olive">c</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> &amp; <font color="Olive">c</font> <a href="normform.html#R1">c=</a> <font color="Olive">b</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</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>,   <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>] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="normform.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> );<br/>
<div><i><font color="Green">E23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><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="E1:33_1"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font>
 ;<br/><b>then consider </b><font color="Maroon">c</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="E3:33_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font> &amp; <font color="Maroon">c</font> <a href="normform.html#R1">c=</a> <font color="Maroon">b</font> <a href="normform.html#K1">\/</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as    <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> ;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">c</font>;<br/><b>thus </b><a NAME="E5:33_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">b</font>,<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E5:33"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a>  <a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b</font>,<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font>]
 <b>from </b><i><a class="ref" href="fraenkel.html#S27">FRAENKEL:sch 27</a>();<br/></i>
<a NAME="E6:33"/>
for <font color="Olive">b</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font> holds <br/><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v</font>
 
<b>by </b><i/>;<br/>
<a NAME="E7:33"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">f'</font> <a href="setwiseo.html#K8">.:</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">v</font>
 
<b>by </b><i><a class="ref" href="setwiseo.html#T15">SETWISEO:15</a></i>;<br/>
<b>set </b><font color="Maroon">d</font> =  <a href="normform.html#K6">FinPairUnion</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span>,<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="Maroon">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><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</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/><a NAME="E1:33_2"/><i><font color="Green">E61</font></i>: 
<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="Maroon">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> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font></span>)</span> <a href="normform.html#K3">\</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>assume </b><a NAME="E2:33_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <font color="Maroon">u</font>
 ;<br/><a NAME="E3:33_2"/><b>then </b>
<font color="Maroon">f'</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font> <a href="normform.html#K1">\/</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:33_2"/>
<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="Maroon">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> <a href="funct_2.html#K3">.</a> <font color="Maroon">s</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="normform.html#T31">NORMFORM:31</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:33"/><b>then </b>
 <a href="normform.html#K6">FinPairUnion</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span>,<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="Maroon">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> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="normform.html#T40">NORMFORM:40</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">d</font> =  <a href="normform.html#K6">FinPairUnion</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span>,<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="Maroon">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> as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="normform.html#T45">NORMFORM:45</a></i>;<br/>
<b>take </b>
<font color="Maroon">d</font>
;<br/>

<a NAME="E11:33"/>
<font color="Maroon">d</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> <span class="p3">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span>,<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> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">v</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E12:33"/><b>then </b>
<font color="Maroon">d</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> <span class="p3">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span>,<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> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">v</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E13:33"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">u</font></span>)</span> <a href="heyting1.html#K7" target="_self">=&gt;&gt;</a> <span class="p1">(<span class="default"><a href="heyting1.html#K3" target="_self">@</a> <font color="Maroon">v</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E14:33"/>
<font color="Maroon">d</font> <a href="normform.html#R1">c=</a> <font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="normform.html#T40">NORMFORM:40</a></i>;<br/>


</div>
