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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K23" target="_self">the_Complex_Space</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="complsp1.html#K8" target="_self">COMPLEX</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:113"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:113"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="complsp1.html#K23" target="_self">the_Complex_Space</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="complsp1.html#E1:113"><i><font color="Green">E88</font></i></a></i>;<br/>
<a NAME="E3:113"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="complsp1.html#K23" target="_self">the_Complex_Space</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> iff <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="complsp1.html#V1" target="_self">open</a> )
 
<b>by </b><i><a class="ref" href="complsp1.html#T108" target="_self">Th108</a></i>;<br/>
<b>hence </b><a NAME="E4:113"/>
( <font color="Maroon">c<sub>3</sub></font> is <a href="complsp1.html#V2" target="_self">closed</a> iff <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="complsp1.html#T109" target="_self">Th109</a>, <a class="ref" href="pre_topc.html#D6">PRE_TOPC:def 6</a></i>;<br/>


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