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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="E1:50"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<b>assume </b><a NAME="E2:50"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 ;<br/>

<a NAME="E3:50"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E4:50"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <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#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<b>hence </b><a NAME="E5:50"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E1:50"><i><font color="Green">E30</font></i></a>, <a class="txt" href="tops_3.html#E3:50"><i><font color="Green">E31</font></i></a>, <a class="ref" href="tops_3.html#T44" target="_self">Th44</a></i>;<br/>


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