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<b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <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> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<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">g</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<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>set </b><font color="Maroon">J</font> =  <a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font>;<br/>
<a NAME="E1:28"/><i><font color="Green">E29</font></i>: 
(  <a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a>  &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="normform.html#T5" target="_self">Th5</a>, <a class="ref" href="normform.html#T10" target="_self">Th10</a></i>;<br/>
<b>assume </b><a NAME="E2:28"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">f</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relat_1.html#K7">|</a> <font color="Maroon">B</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:28_1_1"/>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><b>suppose </b></a><a NAME="E1:28_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b> <a href="normform.html#K6" target="_self">FinPairUnion</a> <font color="Maroon">B</font>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font></span>)</span> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">f</font>

<br/>.= 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">A</font></span>)</span></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="ref" href="normform.html#T12" target="_self">Th12</a>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font></span>)</span> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1">{}.</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="normform.html#T12" target="_self">Th12</a>, <a class="ref" href="setwiseo.html#T40">SETWISEO:40</a></i>
<br/>.= 
 <a href="normform.html#K6" target="_self">FinPairUnion</a> <font color="Maroon">B</font>,<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="normform.html#T20" target="_self">Th20</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><b>suppose </b></a><a NAME="E1:28_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:28_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="setwiseo.html#K8">.:</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="normform.html#T16" target="_self">Th16</a>, <a class="ref" href="lattice2.html#T16">LATTICE2:16</a></i>;<br/><b>thus </b><a NAME="E3:28_1_1_2"/>
 <a href="normform.html#K6" target="_self">FinPairUnion</a> <font color="Maroon">B</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="normform.html#K6" target="_self">FinPairUnion</a> <font color="Maroon">B</font>,<font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="normform.html#T20" target="_self">Th20</a>, <a class="ref" href="normform.html#T21" target="_self">Th21</a>, <a class="ref" href="setwiseo.html#T35">SETWISEO:35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:28"/>
 <a href="normform.html#K6" target="_self">FinPairUnion</a> <font color="Maroon">B</font>,<font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="normform.html#K6" target="_self">FinPairUnion</a> <font color="Maroon">B</font>,<font color="Maroon">g</font>
 ;<br/>


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