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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="pencil_4.html#K8" title="PENCIL_4:func.8">PairSet</a> <font color="Maroon" title="c1">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">H<sub>1</sub></font>()] <b>means </b> ex <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">l</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">l</font> );<br/>
<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon">H<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:42_1_1"/><i><font color="Green" title="E34">A1</font></i>: 
for <font color="Olive" title="b1">B</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon">H<sub>1</sub></font>() holds <br/> ( <font color="Olive" title="b1">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">B</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3" title="SUBSET_1:sch.3">SUBSET_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c3">F</font>
;<br/>

<b>let </b><font color="Maroon" title="c4">S</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E3:42_1_1"/>
( <font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies  ex <font color="Olive" title="b1">t</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ex <font color="Olive" title="b2">l</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Olive" title="b2">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> <font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">l</font> ) )
 <b>by </b><i><a class="txt" href="pencil_4.html#E2:42_1_1"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/>

<b>given </b><font color="Maroon" title="c5">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> , <font color="Maroon" title="c6">l</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><a NAME="E5:42_1_1"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c5">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">L</font> &amp; <font color="Maroon" title="c4">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pencil_4.html#K9" title="PENCIL_4:func.9">PairSet</a> <font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c6">l</font> )
 ;<br/>

<a NAME="E6:42_1_1"/>
<font color="Maroon" title="c4">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pencil_4.html#K8" title="PENCIL_4:func.8">PairSet</a> <font color="Maroon" title="c1">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green" title="E36">A3</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">l</font> &amp; <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c5">t</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="pencil_4.html#E5:42_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="pencil_4.html#D9" target="_self" title="PENCIL_4:def.9">Def9</a></i>;<br/>
<b>thus </b><a NAME="E4:42_1_1_1"/>
<font color="Maroon" title="c7">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pencil_4.html#K8" title="PENCIL_4:func.8">PairSet</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E5:42_1_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="txt" href="pencil_4.html#E3:42_1_1_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="pencil_4.html#D8" target="_self" title="PENCIL_4:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:42_1_1"/>
<font color="Maroon" title="c4">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="pencil_4.html#E2:42_1_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="pencil_4.html#E5:42_1_1"><i><font color="Green" title="E35">A2</font></i></a></i>;<br/>


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