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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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">BB</font> be    <a href="cantor_1.html#M2">prebasis</a> of <font color="Maroon">T</font>;<br/>







<b>assume </b><a NAME="E1:56"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">Z</font> being <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>  st <font color="Olive">Z</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">Z</font> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">Z</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">BB'</font> =  <a href="cantor_1.html#K2">FinMeetCl</a> <font color="Maroon">BB</font> as    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>assume </b><a NAME="E1:56_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">BB'</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="E3:56_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">BB</font> &amp; <font color="Maroon">Y</font> is <a href="finset_1.html#V1">finite</a> &amp; <font color="Maroon">A</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/><b>reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">Y</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><b>assume </b><a NAME="E5:56_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/><b>hence </b><a NAME="E6:56_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">X</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:56"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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