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<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">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">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>;<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">c</font> be    <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/>









<a NAME="E1:27"/><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> 
 
<b>by </b><i><a class="ref" href="normform.html#T5" target="_self">Th5</a></i>;<br/>
<b>assume </b><a NAME="E2:27"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font>] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> implies <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> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">f</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font> );<br/>
<a NAME="E3:27"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">X</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/>
 <a href="setwiseo.html#K1">{}.</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:27_1"/>
 <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> 
 
<b>by </b><i><a class="ref" href="normform.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E3:27_1"/><b>then </b>
<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> <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="setwiseo.html#T40">SETWISEO:40</a></i>;<br/>
<b>hence </b><a NAME="E4:27_1"/>
<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> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="normform.html#T11" target="_self">Th11</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><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> <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">b</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:27_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:27_2_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:27_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">b</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T11">XBOOLE_1:11</a></i>;<br/><a NAME="E3:27_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><i><font color="Green">E50</font></i>: <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">$$</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font></span>)</span> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">C</font>,<font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="normform.html#T12" target="_self">Th12</a>, <a class="ref" href="setwiseo.html#T41">SETWISEO:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font></span>)</span> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">C</font>,<font color="Maroon">f</font></span>)</span> <a href="normform.html#K1" target="_self">\/</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font></span>)</span>
<b>by </b><i/>
;<br/><a NAME="E5:27_2_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">b</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="normform.html#T16" target="_self">Th16</a>, <a class="ref" href="normform.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E6:27_2_1"/>
<span class="p1">(<span class="default"><a href="normform.html#K5" target="_self">FinPairUnion</a> <font color="Maroon">A</font></span>)</span> <a href="setwiseo.html#K7">$$</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="setwiseo.html#K5">\/</a> <span class="p2"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span></span>)</span>,<font color="Maroon">f</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="normform.html#T22" target="_self">Th22</a>, <a class="ref" href="normform.html#T25" target="_self">Th25</a>, <a class="ref" href="normform.html#T28" target="_self">Th28</a>, , <a class="ref" href="xboole_1.html#T11">XBOOLE_1:11</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:27_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">C</font> <a href="setwiseo.html#K5">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><font color="Maroon">b</font></span><a href="setwiseo.html#K2">.}</a></span>]
 ;<br/></div><b>end;</b></div>
<a NAME="E5:27"/>
for <font color="Olive">C</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">C</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4">SETWISEO:sch 4</a>(<a class="ref" href="normform.html#T20" target="_self">Th20</a>, <a class="ref" href="normform.html#T21" target="_self">Th21</a>);<br/></i>
<a NAME="E6:27"/><b>then </b>
<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> <font color="Maroon">B</font>,<font color="Maroon">f</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font>
 
;<br/>
<b>hence </b><a NAME="E7:27"/>
 <a href="normform.html#K6" target="_self">FinPairUnion</a> <font color="Maroon">B</font>,<font color="Maroon">f</font> <a href="normform.html#R1" target="_self">c=</a> <font color="Maroon">c</font>
 ;<br/>


</div>
