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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:38"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="ref" href="cantor_1.html#D4">CANTOR_1:def 4</a></i>;<br/><a NAME="E4:38_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> holds <br/><font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/><a NAME="E5:38_1"/>
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">Y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E6:38_1"/><b>then </b>
( ( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> ) or <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/><b>hence </b><a NAME="E7:38_1"/>
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="yellow19.html#T2" target="_self">Th2</a>, <a class="ref" href="pre_topc.html#T44">PRE_TOPC:44</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:38"/>
 <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">F</font> is <a href="tops_2.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>


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