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

<b>let </b><font color="Maroon">X0</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:21_2_1"/><i><font color="Green">E24</font></i>: 
not <font color="Maroon">X0</font> is <a href="tex_3.html#V1" target="_self">dense</a>
 ;<br/>

<b>assume </b><a NAME="E2:21_2_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X0</font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:21_2_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font>
 ;<br/><a NAME="E2:21_2_1_1"/><b>then </b>
<font color="Maroon">A</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:21_2_1_1"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T33">TOPS_3:33</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:21_2_1"/>
contradiction
 <b>by </b><i>, </i>;<br/>


</div>
