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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span>}</span>  &amp; <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( <font color="Maroon">P<sub>2</sub></font>[$1] &amp; <font color="Maroon">P<sub>1</sub></font>[$1] );<br/>
<a NAME="E1:17_1"/><i><font color="Green" title="E4">A1</font></i>: 
for <font color="Olive" title="b1">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">s</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">s</font>] )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon" title="c1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">s1</font> where <font color="Olive" title="b1">s1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">s</font>]</span><br/><a NAME="E2:17_1_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">s1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon" title="c1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s1</font> &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Olive" title="b1">s1</font>] )
 ;<br/><b>hence </b><a NAME="E3:17_1_1_1"/>
<font color="Maroon">P<sub>2</sub></font>[<font color="Maroon" title="c1">s</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:17_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">s</font>] iff <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c1">s</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:17_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">s</font>) where <font color="Olive" title="b1">s</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">s2</font>) where <font color="Olive" title="b1">s2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">s2</font>] </span>}</span> 
 <b>from </b><i><a class="ref" href="fraenkel.html#S3" target="_self" title="FRAENKEL:sch.3">FRAENKEL:sch 3</a>(<a class="txt" href="fraenkel.html#E1:17_1"><i><font color="Green" title="E4">A1</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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