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<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> ;<br/><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>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>





<b>thus </b><a NAME="E1:50"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font> implies ( not <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a> &amp; ( for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3" target="_self">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font> ) ) )
 <b>by </b><i>, <a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>

<b>assume </b><a NAME="E2:50"/>
not <font color="Maroon">T</font> is <a href="struct_0.html#V3">empty</a>
 ;<br/>

<a NAME="E3:50"/><b>then </b><i><font color="Green">E19</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="struct_0.html#D1">STRUCT_0:def 1</a></i>;<br/>
<b>assume </b><a NAME="E4:50"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">G</font> is <a href="pre_topc.html#V3" target="_self">open</a> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">G</font> holds <br/><font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 ;<br/>

<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4" target="_self">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


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