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<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <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>] <span class="kw">means </span>( <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> &amp; <font color="Maroon" title="c1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  );<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">Z</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">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>] </span> } </span>  as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <span class="kw">from</span> <span class="lab"><a class="ref" href="domain_1.html#S7" onmouseover="rs('domain_1/S7')" onmouseout="rh()">DOMAIN_1:sch 7</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c3">Z</font> = <font color="Maroon" title="c2">Z</font> as   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>
<a NAME="E3:75_1_1"/>
<font color="Maroon" title="c3">Z</font> is  <a href="measure6.html#V2" title="MEASURE6:attr.2">closed</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">Y</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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="measure6.html#D7" target="_self" onmouseover="rs('measure6/D7')" onmouseout="rh()">MEASURE6:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Z</font> implies <font color="Maroon" title="c4">Y</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:75_1_1_1"/>
<font color="Maroon" title="c4">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">Z</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">Y</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> </span><br/>

<a NAME="E2:75_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">A</font> being   <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> st <br/>( <font color="Maroon" title="c4">Y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> &amp; <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>  )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:75_1_1_1"/>
<font color="Maroon" title="c4">Y</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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">hence </span><a NAME="E4:75_1_1"/>
 <a href="measure6.html#K5" title="MEASURE6:func.5">Cl</a> <font color="Maroon" title="c1">X</font> is  <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="measure6.html#T56" target="_self" onmouseover="rs('measure6/T56')" onmouseout="rh()">Th56</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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