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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/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> is <a href="rcomp_1.html#V2">closed</a>
 ;<br/><a NAME="E2:55_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">X</font> <a href="tarski.html#R1">c=</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<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="E3:55_1"/>
<font color="Maroon">X</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> 
 <b>by </b><i/>;<br/><a NAME="E4:55_1"/><b>then </b>
 <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4">SETFAM_1:4</a></i>;<br/><b>hence </b><a NAME="E5:55_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:55"/>
( <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K6" target="_self">Cl</a> <font color="Maroon">X</font> implies <font color="Maroon">X</font> is <a href="rcomp_1.html#V2">closed</a> )
 ;<br/>


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