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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <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/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:63_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="topgen_1.html#V5" target="_self">perfect</a>
 ;<br/><a NAME="E2:63_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="topgen_1.html#D10" target="_self">Def10</a></i>;<br/><b>hence </b><a NAME="E3:63_1"/>
 <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="topgen_1.html#E54"><i><font color="Green">Lemma36</font></i></a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:63"/><i><font color="Green">E38</font></i>: 
 <a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E3:63"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="topgen_1.html#V2" target="_self">dense-in-itself</a>
 
<b>by </b><i><a class="ref" href="topgen_1.html#D7" target="_self">Def7</a></i>;<br/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>2</sub></font> = 
<span class="p1">(<span class="default"><a href="topgen_1.html#K2" target="_self">Der</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="topgen_1.html#T31" target="_self">Th31</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="topgen_1.html#E2:63"><i><font color="Green">E38</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E5:63"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="topgen_1.html#V5" target="_self">perfect</a>
 <b>by </b><i><a class="txt" href="topgen_1.html#E3:63"><i><font color="Green">E39</font></i></a>, <a class="ref" href="topgen_1.html#D10" target="_self">Def10</a></i>;<br/>


</div>
