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<a NAME="E1:1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">F<sub>1</sub></font>() is <a href="finset_1.html#V1">finite</a>
 
;<br/>
<b>set </b><font color="Maroon">B</font> = <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> ;<br/>
<a NAME="E2:1_1"/><i><font color="Green">E31</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>();<br/></i>
<a NAME="E3:1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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="E1:1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</font>] </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon">b</font>) <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E4:1_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">b</font>]
 ;<br/>
<b>thus </b><a NAME="E5:1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>1</sub></font>() </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">x</font>) where <font color="Olive">x</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">x</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/>


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