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<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>assume </b><a NAME="E1:39_4_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V3" target="_self">boundary</a>
 ;<br/>

<b>assume </b><a NAME="E2:39_4_1"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tex_3.html#V2" target="_self">everywhere_dense</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="tsep_1.html#T1">TSEP_1:1</a></i>;<br/>
<a NAME="E4:39_4_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tex_3.html#E2:39_4_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tex_3.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E5:39_4_1"/><b>then </b>
( not <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V2">boundary</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V2">boundary</a> )
 
<b>by </b><i><a class="txt" href="tex_3.html#E1:39_4_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="tex_3.html#D3" target="_self">Def3</a>, <a class="ref" href="tops_3.html#T37">TOPS_3:37</a></i>;<br/>
<b>hence </b><a NAME="E6:39_4_1"/>
contradiction
 ;<br/>


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