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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="E1:46"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a> )
 ;<br/>

<a NAME="E2:46"/><b>then </b><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V3">nowhere_dense</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V3">nowhere_dense</a> )
 
<b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self">Th39</a></i>;<br/>
<a NAME="E3:46"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>;<br/>
<a NAME="E4:46"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span> is <a href="tops_1.html#V3">nowhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E2:46"><i><font color="Green">E29</font></i></a>, <a class="ref" href="tops_1.html#T90">TOPS_1:90</a></i>;<br/>
<b>hence </b><a NAME="E5:46"/>
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</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#E3:46"><i><font color="Green">E30</font></i></a>, <a class="ref" href="tops_3.html#T39" target="_self">Th39</a></i>;<br/>


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