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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  iff <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c1">X</font> )</span><br/>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K6" title="MEASURE6: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>  )</span>
<div class="add"><span class="kw">set </span><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/><span class="kw">assume </span><a NAME="E1:79_1"/>
<font color="Maroon" title="c1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c1">X</font></span><br/><a NAME="E2:79_1"/><span class="kw">then </span>
<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> 
 ;<br/><a NAME="E3:79_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <a href="measure6.html#K6" title="MEASURE6: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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#T3" title="SETFAM_1:th.3">SETFAM_1:3</a></span>;<br/><a NAME="E4:79_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T58" target="_self" title="MEASURE6:th.58">Th94</a></span>;<br/><span class="kw">hence </span><a NAME="E5:79_1"/>
<font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K6" title="MEASURE6:func.6">Cl</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:79_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:79"/>
( <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#K6" title="MEASURE6: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>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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