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<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="E1:39_3_1"/><i><font color="Green">E36</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="39_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_3_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X0</font> is <a href="tex_3.html#V3" target="_self">boundary</a>
 ;<br/><b>reconsider </b><font color="Maroon">A</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/><a NAME="E3:39_3_1_1"/>
<font color="Maroon">A</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E2"><i><font color="Green">Lemma17</font></i></a>, </i>;<br/><a NAME="E4:39_3_1_1"/><b>then </b>
( not <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a> &amp; <font color="Maroon">A</font> is <a href="tops_1.html#V2">boundary</a> )
 <b>by </b><i><a class="txt" href="tex_3.html#E3"><i><font color="Green">Lemma18</font></i></a>, <a class="ref" href="tex_3.html#T1" target="_self">Th1</a>, <a class="ref" href="tops_3.html#T37">TOPS_3:37</a></i>;<br/><b>hence </b><a NAME="E5:39_3_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:39_3_1"/>
not <font color="Maroon">X0</font> is <a href="tex_3.html#V3" target="_self">boundary</a>
 ;<br/>


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