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

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



<b>thus </b><a NAME="E1:18"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V2">boundary</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1">dense</a>
 
;<br/>
<b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V2">boundary</a>
 <b>by </b><i><a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a></i>;<br/>


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

<b>assume </b><a NAME="E2:18"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<a NAME="E3:18"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <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/>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>


</div>
