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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>() <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  =  <a href="pencil_4.html#K8" target="_self">PairSet</a> <font color="Maroon">X</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">H<sub>1</sub></font>()] <b>means </b> ex <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">l</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K9" target="_self">PairSet</a> <font color="Olive">t</font>,<font color="Olive">l</font> );<br/>
<b>consider </b><font color="Maroon">F</font> being   <a href="setfam_1.html#NM1">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">E24</font></i>: 
for <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">H<sub>1</sub></font>() holds <br/> ( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">B</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>let </b><font color="Maroon">S</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>thus </b><a NAME="E3:42_1_1"/>
( <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies  ex <font color="Olive">t</font> being   <a href="hidden.html#M1">set</a>  ex <font color="Olive">l</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Olive">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K9" target="_self">PairSet</a> <font color="Olive">t</font>,<font color="Olive">l</font> ) )
 <b>by </b><i/>;<br/>

<b>given </b><font color="Maroon">t</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">l</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E4:42_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">l</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="pencil_4.html#K9" target="_self">PairSet</a> <font color="Maroon">t</font>,<font color="Maroon">l</font> )
 ;<br/>

<a NAME="E5:42_1_1"/>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="pencil_4.html#K8" target="_self">PairSet</a> <font color="Maroon">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">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

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

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:42_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">l</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">t</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E4:42_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pencil_4.html#K8" target="_self">PairSet</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="pencil_4.html#T3" target="_self">Th3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:42_1_1"/>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


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