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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>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:7"/><i><font color="Green">E7</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>
 ;<br/>

<b>assume </b><a NAME="E2:7"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V2">boundary</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="tmap_1.html#K6">modified_with_respect_to</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> <b>by </b><i><a class="ref" href="tmap_1.html#T104">TMAP_1:104</a></i>;<br/>
<a NAME="E4:7"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V2">boundary</a> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="txt" href="tex_1.html#E2:7"><i><font color="Green">E8</font></i></a>, <a class="ref" href="tex_1.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:7"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="tex_1.html#E1:7"><i><font color="Green">E7</font></i></a>, <a class="ref" href="tops_1.html#T93">TOPS_1:93</a></i>;<br/>
<b>hence </b><a NAME="E6:7"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V3">nowhere_dense</a>
 <b>by </b><i><a class="ref" href="tops_3.html#T26">TOPS_3:26</a></i>;<br/>


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