<?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="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:45"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>hence </b><a NAME="E2:45"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a> )
 <b>by </b><i><a class="ref" href="tex_3.html#D4" target="_self">Def4</a></i>;<br/>

<b>assume </b><a NAME="E3:45"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/>

<a NAME="E4:45"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 
<b>by </b><i><a class="txt" href="tex_3.html#E1:45"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:45"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V4" target="_self">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tex_3.html#D4" target="_self">Def4</a></i>;<br/>


</div>
