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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>assume </b><a NAME="E1:51"/>
<font color="Maroon" title="c2">D</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">C</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:51"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font>
 <b>and </b><br/><a NAME="E4:51"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:51"/><i><font color="Green" title="E32">A3</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T41" target="_self" title="TOPS_3:th.41">Th41</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">C</font>
;<br/>

<b>take </b><font color="Maroon" title="c4">B</font> = <font color="Maroon" title="c2">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font>;<br/>

<b>thus </b><a NAME="E6:51"/>
( <font color="Maroon" title="c3">C</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">C</font> is <a href="tops_1.html#V1" title="TOPS_1:attr.1">dense</a> )
 <b>by </b><i><a class="txt" href="tops_3.html#E4:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="tops_3.html#E5:51"><i><font color="Green" title="E32">A3</font></i></a></i>;<br/>

<a NAME="E7:51"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></i>;<br/>
<a NAME="E8:51"/>
<font color="Maroon" title="c3">C</font> is <a href="tops_3.html#V1" title="TOPS_3:attr.1">everywhere_dense</a>
 
<b>by </b><i><a class="txt" href="tops_3.html#E4:51"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="tops_3.html#E5:51"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="tops_3.html#T36" target="_self" title="TOPS_3:th.36">Th36</a></i>;<br/>
<a NAME="E9:51"/><b>then </b>
<font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 
<b>by </b><i><a class="ref" href="tops_3.html#T39" target="_self" title="TOPS_3:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E10:51"/>
<font color="Maroon" title="c4">B</font> is <a href="tops_1.html#V3" title="TOPS_1:attr.3">nowhere_dense</a>
 <b>by </b><i><a class="txt" href="tops_3.html#E7:51"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="tops_3.html#T26" target="_self" title="TOPS_3:th.26">Th26</a></i>;<br/>

<b>thus </b><a NAME="E11:51"/>
( <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> &amp; <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">B</font> )
 <b>by </b><i><a class="txt" href="tops_3.html#E3:51"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a>, <a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>


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