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

<b>let </b><font color="Maroon" title="c2">X0</font> be    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c3">A</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">X0</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1" title="TSEP_1:th.1">TSEP_1:1</a></i>;<br/>
<b>assume </b><a NAME="E2:73_3_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">X0</font> is <a href="tex_3.html#V2" title="TEX_3:attr.2">everywhere_dense</a>
 ;<br/>

<b>assume </b><a NAME="E3:73_3_1"/>
<font color="Maroon" title="c2">X0</font> is <a href="tex_2.html#V2" title="TEX_2:attr.2">proper</a>
 ;<br/>

<a NAME="E4:73_3_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> is <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a>
 
<b>by </b><i><a class="ref" href="tex_2.html#T13" title="TEX_2:th.13">TEX_2:13</a></i>;<br/>
<a NAME="E5:73_3_1"/><b>then </b>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>
 
<b>by </b><i><a class="ref" href="tex_2.html#T5" title="TEX_2:th.5">TEX_2:5</a></i>;<br/>
<a NAME="E6:73_3_1"/><b>then </b>
not <font color="Maroon" title="c3">A</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tex_1.html#T36" title="TEX_1:th.36">TEX_1:36</a></i>;<br/>
<b>hence </b><a NAME="E7:73_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="tex_3.html#E2:73_3_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="tex_3.html#T16" target="_self" title="TEX_3:th.16">Th16</a></i>;<br/>


</div>
