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

<b>let </b><font color="Maroon">T</font> be    <a href="pre_topc.html#L1">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>thus </b><a NAME="E1:6"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> implies <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:6_1"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:6_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</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">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<b>hence </b><a NAME="E3:6_1"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:6"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E3:6"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</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">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>
<b>hence </b><a NAME="E4:6"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>


</div>
