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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> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:51"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_3.html#V1" target="_self">everywhere_dense</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:51"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:51"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:51"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T41" target="_self">Th41</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>3</sub></font>
;<br/>

<b>take </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font>;<br/>

<b>thus </b><a NAME="E6:51"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V1">dense</a> )
 <b>by </b><i><a class="txt" href="tops_3.html#E4:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tops_3.html#E5:51"><i><font color="Green">E32</font></i></a></i>;<br/>

<a NAME="E7:51"/>
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E8:51"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 
;<br/>
<a NAME="E9:51"/>
<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#E4:51"><i><font color="Green">E31</font></i></a>, <a class="txt" href="tops_3.html#E5:51"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_3.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E10:51"/><b>then </b>
<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/>
<b>hence </b><a NAME="E11:51"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E8:51"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tops_3.html#T26" target="_self">Th26</a></i>;<br/>

<b>thus </b><a NAME="E12:51"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="tops_3.html#E3:51"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a>, <a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


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