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

<b>let </b><font color="Maroon">X</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">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:6"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:6"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<b>assume </b><a NAME="E3:6"/>
<font color="Maroon">D</font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<a NAME="E4:6"/><b>then </b>
<font color="Maroon">D</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a></i>;<br/>
<a NAME="E5:6"/><b>then </b>
( <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1">dense</a> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:6"/>
( <font color="Maroon">C</font> is <a href="tops_1.html#V2">boundary</a> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a>, <a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a></i>;<br/>


</div>
