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<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>, <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="E1:4"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2">constitute_a_decomposition</a> 
 ;<br/>

<a NAME="E2:4"/><b>then </b><i><font color="Green">E5</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="tsep_2.html#T4">TSEP_2:4</a></i>;<br/>
<b>hence </b><a NAME="E3:4"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a> implies <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V2">boundary</a> )
 <b>by </b><i><a class="ref" href="tops_3.html#T18">TOPS_3:18</a></i>;<br/>

<b>assume </b><a NAME="E4:4"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<b>hence </b><a NAME="E5:4"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="txt" href="tex_3.html#E2:4"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tops_1.html#D4">TOPS_1:def 4</a></i>;<br/>


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