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<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">P</font> is <a href="tops_1.html#V3" target="_self">nowhere_dense</a>
 <b>and </b><br/><a NAME="E3:97"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E4:97"/>
<font color="Maroon">P</font> <a href="subset_1.html#K3">`</a>  is <a href="tops_1.html#V1" target="_self">dense</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T54" target="_self">Th54</a>, <a class="ref" href="tops_1.html#T46" target="_self">Th46</a></i>;<br/>
<a NAME="E5:97"/><b>then </b>
<font color="Maroon">P</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">P</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55" target="_self">Th55</a>, </i>;<br/>
<b>hence </b><a NAME="E6:97"/>
contradiction
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>


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