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<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><b>let </b><font color="Maroon" title="c4">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">X</font>;<br/><b>let </b><font color="Maroon" title="c5">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>









<b>assume </b><a NAME="E1:27"/><i><font color="Green" title="E18">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font> holds <br/><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">X</font>] <b>means </b>( $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font> implies <span class="p1">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> $1,<font color="Maroon" title="c3">f</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font> );<br/>
<a NAME="E2:27"/><i><font color="Green" title="E19">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">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" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 ;<br/>

<a NAME="E2:27_1"/>
<span class="p1">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binop_1.html#K3" title="BINOP_1:func.3">the_unity_wrt</a> <span class="p1">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="normform.html#T37" target="_self" title="NORMFORM:th.37">Th37</a>, <a class="ref" href="setwiseo.html#T40" title="SETWISEO:th.40">SETWISEO:40</a></i>;<br/>
<b>hence </b><a NAME="E3:27_1"/>
<span class="p1">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c3">f</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="ref" href="normform.html#T39" target="_self" title="NORMFORM:th.39">Th39</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E20">A3</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" title="c6">C</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">X</font>;<br/><b>let </b><font color="Maroon" title="c7">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>;<br/><b>assume </b><a NAME="E1:27_2"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">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" title="E22">A5</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 ;<br/><a NAME="E2:27_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">XBOOLE_1:11</a></i>;<br/><a NAME="E3:27_2_1"/><b>then </b><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">B</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><i><font color="Green" title="E24">A7</font></i>: <span class="p1">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c3">f</font> = 
<span class="p1">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c3">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
<b>by </b><i><a class="ref" href="normform.html#T37" target="_self" title="NORMFORM:th.37">Th37</a>, <a class="ref" href="setwiseo.html#T41" title="SETWISEO:th.41">SETWISEO:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <font color="Maroon" title="c6">C</font>,<font color="Maroon" title="c3">f</font></span>)</span> <a href="normform.html#K1" title="NORMFORM:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font></span>)</span>
<b>by </b><i><a class="ref" href="normform.html#D6" target="_self" title="NORMFORM:def.6">Def6</a></i>
;<br/><a NAME="E5:27_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">b</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="normform.html#E1:27"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="normform.html#E3:27_2_1"><i><font color="Green" title="E23">A6</font></i></a></i>;<br/><b>hence </b><a NAME="E6:27_2_1"/>
<span class="p1">(<span class="default"><a href="normform.html#K5" title="NORMFORM:func.5">FinPairUnion</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">C</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p2"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c3">f</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="normform.html#E1:27_2"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="normform.html#E1:27_2_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="normform.html#E4:27_2_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="normform.html#T25" target="_self" title="NORMFORM:th.25">Th25</a>, <a class="ref" href="xboole_1.html#T11" title="XBOOLE_1:th.11">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" title="c6">C</font> <a href="setwiseo.html#K5" title="SETWISEO:func.5">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c7">b</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>]
 ;<br/></div><b>end;</b></div>
<a NAME="E4:27"/>
for <font color="Olive" title="b1">C</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">X</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">C</font>]
 
<b>from </b><i><a class="ref" href="setwiseo.html#S4" title="SETWISEO:sch.4">SETWISEO:sch 4</a>(<a class="txt" href="normform.html#E2:27"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="normform.html#E3:27"><i><font color="Green" title="E20">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:27"/>
 <a href="normform.html#K6" title="NORMFORM:func.6">FinPairUnion</a> <font color="Maroon" title="c4">B</font>,<font color="Maroon" title="c3">f</font> <a href="normform.html#R1" title="NORMFORM:pred.1">c=</a> <font color="Maroon" title="c5">c</font>
 ;<br/>


</div>
