<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="pre_topc.html#L1" target="_self">TopStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:50"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E2:50"/><b>then </b><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3" target="_self">open</a> iff <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4" target="_self">closed</a> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6" target="_self">Def6</a></i>;<br/>
<b>hence </b><a NAME="E3:50"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3" target="_self">open</a> implies  <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T52" target="_self">Th52</a></i>;<br/>

<b>assume </b><a NAME="E4:50"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V2" target="_self">TopSpace-like</a> &amp;  <a href="pre_topc.html#K4" target="_self">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2" target="_self">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>

<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3" target="_self">open</a>
 <b>by </b><i><a class="txt" href="pre_topc.html#E2:50"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T52" target="_self">Th52</a></i>;<br/>


</div>
