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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_1"/><i><font color="Green" title="E33">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 ;<br/><a NAME="E2:54_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T33" target="_self" title="PSCOMP_1:th.33">Th33</a></i>;<br/><b>set </b><font color="Maroon" title="c2">ClX</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> ) </span>}</span> ;<br/><a NAME="E3:54_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">A</font> where <font color="Olive" title="b1">A</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  : ( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b1">A</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:54_1"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/><a NAME="E4:54_1"/><b>then </b>
 <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T4" title="SETFAM_1:th.4">SETFAM_1:4</a></i>;<br/><b>hence </b><a NAME="E5:54_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:54_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:54"/>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">Cl</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c1">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> )
 ;<br/>


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