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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s1</font> where <font color="Olive">s1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">s1</font>] </span>}</span>  &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>] );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>] );<br/>
<a NAME="E1:18_1"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">t</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></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>2</sub></font>();<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">s1</font> where <font color="Olive">s1</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">s1</font>] </span>}</span> 
 ;<br/><a NAME="E2:18_1_1_1"/><b>then </b>
 ex <font color="Olive">s1</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">s1</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive">s1</font>] )
 ;<br/><b>hence </b><a NAME="E3:18_1_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">s</font>]
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:18_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">t</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">s</font>,<font color="Maroon">t</font>] )
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:18_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">s</font>,<font color="Olive">t</font>) where <font color="Olive">s</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <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="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>,<font color="Olive">t</font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">s2</font>,<font color="Olive">t2</font>) where <font color="Olive">s2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive">t2</font> is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">s2</font>,<font color="Olive">t2</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S4" target="_self">FRAENKEL:sch 4</a>();</i><br/>


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