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

<b>set </b><font color="Maroon">N</font> = <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/>
<b>set </b><font color="Maroon">IT</font> = <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="E1:21_2_1"/><i><font color="Green">E73</font></i>: 
<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>  <a href="tarski.html#R1">c=</a>  <a href="normform.html#K7">DISJOINT_PAIRS</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E2:21_2_1"/><i><font color="Green">E74</font></i>: 
<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>  <a href="tarski.html#R1">c=</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_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E3:21_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">B</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E4:21_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">C</font> is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>-&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> =  <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="Maroon">a<sub>1</sub></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">E95</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">f</font> be    <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>;<br/><b>let </b><font color="Maroon">f'</font> be    <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>;<br/><b>assume </b><a NAME="E1:21_2_1_1"/>
<font color="Maroon">f</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="partfun1.html#K2">|</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:21_2_1_1"/><b>then </b>
<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="partfun1.html#K2">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <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="partfun1.html#K2">|</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T29">FUNCOP_1:29</a></i>;<br/><b>hence </b><a NAME="E3:21_2_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">f</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">f'</font>)
 <b>by </b><i><a class="ref" href="normform.html#T41">NORMFORM:41</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:21_2_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">f</font>) 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>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S25">FRAENKEL:sch 25</a>(, , );<br/></i>
<a NAME="E7:21_2_1"/><b>then </b>
<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>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<b>hence </b><a NAME="E8:21_2_1"/>
<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>  is    <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>
 <b>by </b><i>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>


</div>
