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<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="tex_2.html#V1">proper</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K6" target="_self">-PointClTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>





<a NAME="E1:57"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> iff <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="ref" href="tops_1.html#T30">TOPS_1:30</a></i>;<br/>
<a NAME="E2:57"/><b>then </b>
( ( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  ) &amp; ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K3">`</a>  implies <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> ) &amp; <font color="Maroon">c<sub>3</sub></font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="topgen_3.html#K6" target="_self">-PointClTop</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="topgen_3.html#D7" target="_self">Def7</a>, <a class="ref" href="topgen_3.html#T38" target="_self">Th38</a></i>;<br/>
<b>hence </b><a NAME="E3:57"/>
( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V3">open</a> iff not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="fin_topo.html#T21">FIN_TOPO:21</a></i>;<br/>


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