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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ] <b>means </b>( <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> is <a href="rcomp_1.html#V2">closed</a> );<br/>
<b>reconsider </b><font color="Maroon">Z</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a>  <b>from </b><i><a class="ref" href="domain_1.html#S7">DOMAIN_1:sch 7</a>();<br/></i>
<b>reconsider </b><font color="Maroon">Z</font> = <font color="Maroon">Z</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<a NAME="E3:52_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">Z</font> is <a href="pscomp_1.html#V4" target="_self">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">Y</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pscomp_1.html#D6" target="_self">PSCOMP_1:def 6</a><br/>

<b>assume </b><a NAME="E1:52_1_1_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font>
 ;<br/>

<a NAME="E2:52_1_1_1"/><b>then </b>
 ex <font color="Olive">A</font> being  <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> is <a href="rcomp_1.html#V2">closed</a> )
 
;<br/>
<b>hence </b><a NAME="E3:52_1_1_1"/>
<font color="Maroon">Y</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:52_1_1"/>
 <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font> is <a href="rcomp_1.html#V2">closed</a>
 <b>by </b><i>, <a class="txt" href="pscomp_1.html#E3:52_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>


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