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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <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>means </b>for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> iff ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">a<sub>1</sub></font>] ) );<br/>
<div><i><font color="Green">E14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <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> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">x</font>] );<br/><b>consider </b><font color="Maroon">y</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E2:16_1_1"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">a</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> iff <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">a</font>] )
 <b>from </b><i><a class="ref" href="subset_1.html#S3">SUBSET_1:sch 3</a>();<br/></i><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">F<sub>2</sub></font>() as   <a href="subset_1.html#NM2">Subset</a> of <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/><a NAME="E4:16_1_1"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">y</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="setwiseo.html#E2:16_1_1"><i><font color="Green">E15</font></i></a></i>;<br/><a NAME="E5:16_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 <b>by </b><i><a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/><a NAME="E6:16_1_1"/><b>then </b>
<font color="Maroon">y</font> 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/><b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> 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">y'</font> = <font color="Maroon">y</font>;<br/><b>thus </b><a NAME="E8:16_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y'</font>]
 <b>by </b><i><a class="txt" href="setwiseo.html#E2:16_1_1"><i><font color="Green">E15</font></i></a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E2:16_1"/>
 ex <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">F<sub>1</sub></font>(), <a href="finsub_1.html#K5">Fin</a> <font color="Maroon">F<sub>1</sub></font>() st <br/>for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="setwiseo.html#E1:16_1"><i><font color="Green">E14</font></i></a>);</i><br/>


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