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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>( <font color="Maroon">P<sub>2</sub></font>[$1] &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() );<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>1</sub></font>[$1] &amp; $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <font color="Maroon">F<sub>4</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1;<br/>
<b>set </b><font color="Maroon" title="c1">C</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v'</font>) where <font color="Olive" title="b1">v'</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">v'</font>] </span>}</span> ;<br/>
<a NAME="E1:12_1"/><i><font color="Green" title="E6">A3</font></i>: 
for <font color="Olive" title="b1">v</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>2</sub></font>[<font color="Olive" title="b1">v</font>] iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">v</font>] )
 
<b>by </b><i><a class="txt" href="fraenkel.html#E2:12"><i><font color="Green" title="E5">A2</font></i></a></i>;<br/>
<a NAME="E2:12_1"/><i><font color="Green" title="E7">A4</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v'</font>) where <font color="Olive" title="b1">v'</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">v'</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v'</font>) where <font color="Olive" title="b1">v'</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">v'</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#E1:12_1"><i><font color="Green" title="E6">A3</font></i></a>);<br/></i>
<a NAME="E3:12_1"/><i><font color="Green" title="E8">A5</font></i>: 
for <font color="Olive" title="b1">v</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 <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">v</font>] holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">v</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v</font>)
 
<b>by </b><i><a class="txt" href="fraenkel.html#E1:12"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fraenkel.html#T3" target="_self" title="FRAENKEL:th.3">Th3</a></i>;<br/>
<a NAME="E4:12_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">u'</font>) where <font color="Olive" title="b1">u'</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">u'</font>] </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">v'</font>) where <font color="Olive" title="b1">v'</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">v'</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S6" target="_self" title="FRAENKEL:sch.6">FRAENKEL:sch 6</a>(<a class="txt" href="fraenkel.html#E3:12_1"><i><font color="Green" title="E8">A5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:12_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>4</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">u'</font></span>)</span> where <font color="Olive" title="b1">u'</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>1</sub></font>[<font color="Olive" title="b1">u'</font>] &amp; <font color="Olive" title="b1">u'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">v'</font></span>)</span> where <font color="Olive" title="b1">v'</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">v'</font>] &amp; <font color="Olive" title="b1">v'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>3</sub></font>() ) </span>}</span> 
 <b>by </b><i><a class="txt" href="fraenkel.html#E2:12_1"><i><font color="Green" title="E7">A4</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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