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<b>let </b><font color="Maroon">X</font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<b>set </b><font color="Maroon">ClX</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">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> ) </span>}</span> ;<br/>
<a NAME="E2:54"/>
 <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#D4">SUBSET_1:def 4</a></i>;<br/>
<a NAME="E3:54"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="subset_1.html#K2">[#]</a> <a href="numbers.html#K1">REAL</a>  <a href="hidden.html#R2">in</a> <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">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> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="fcont_3.html#T1">FCONT_3:1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1"/>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <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">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> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">YY</font> being   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:54_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">YY</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">YY</font> &amp; <font color="Maroon">YY</font> is <a href="rcomp_1.html#V2">closed</a> )
 ;<br/><b>thus </b><a NAME="E4:54_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:54"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/>


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