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<b>let </b><font color="Maroon">X</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">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:5"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 ;<br/>

<b>assume </b><a NAME="E2:5"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">D</font> is <a href="tops_1.html#V1">dense</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <font color="Maroon">D</font></span>)</span> <b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<a NAME="E4:5"/>
( <font color="Maroon">E</font> is <a href="tops_1.html#V1">dense</a> &amp; <font color="Maroon">E</font> is <a href="pre_topc.html#V3">open</a> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:5"/><b>then </b>
( <font color="Maroon">E</font> is <a href="tops_3.html#V1">everywhere_dense</a> &amp; <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="tops_3.html#T36">TOPS_3:36</a></i>;<br/>
<b>hence </b><a NAME="E6:5"/>
<font color="Maroon">C</font> is <a href="tops_3.html#V1">everywhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T38">TOPS_3:38</a></i>;<br/>


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