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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">t'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Olive">t'</font>] );<br/>
<b>set </b><font color="Maroon">c</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) where <font color="Olive">t'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c1</font> = <span class="p1">{<span class="default"> <font color="Olive">tt</font> where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">tt</font>] </span>}</span> ;<br/>
<a NAME="E1:38_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">F<sub>3</sub></font>() is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<a NAME="E2:38_1"/><i><font color="Green">E18</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) where <font color="Olive">t'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21" target="_self">FRAENKEL:sch 21</a>();<br/></i>
<a NAME="E3:38_1"/>
<span class="p1">{<span class="default"> <font color="Olive">tt</font> where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">tt</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) where <font color="Olive">t'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:38_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">tt</font> where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">tt</font>] </span>}</span> 
 ;<br/>

<a NAME="E2:38_1_1"/><b>then </b>
 ex <font color="Olive">tt</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">tt</font> &amp;  ex <font color="Olive">t'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">tt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">tt</font>,<font color="Olive">t'</font>] ) )
 
;<br/>
<b>hence </b><a NAME="E3:38_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) where <font color="Olive">t'</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:38_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">tt</font> where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">tt</font>] </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>
<a NAME="E5:38_1"/>
<span class="p1">{<span class="default"> <font color="Olive">tt</font> where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">tt</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>from </b><i><a class="ref" href="fraenkel.html#S10" target="_self">FRAENKEL:sch 10</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">c1</font> = <span class="p1">{<span class="default"> <font color="Olive">tt</font> where <font color="Olive">tt</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">tt</font>] </span>}</span>  as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i>, <a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">c1</font>
;<br/>

<b>let </b><font color="Maroon">t</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<a NAME="E7:38_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font> implies  ex <font color="Olive">tt</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Olive">tt</font> &amp;  ex <font color="Olive">t'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Olive">tt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">tt</font>,<font color="Olive">t'</font>] ) ) )
 
;<br/>
<b>hence </b><a NAME="E8:38_1"/>
( <font color="Maroon">t</font> <a href="hidden.html#R2">in</a> <font color="Maroon">c1</font> iff  ex <font color="Olive">t'</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() st <br/>( <font color="Olive">t'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>3</sub></font>() &amp; <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">t'</font>) &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">t</font>,<font color="Olive">t'</font>] ) )
 ;<br/>


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