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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>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:13"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tsep_2.html#R2" target="_self">constitute_a_decomposition</a> 
 ;<br/>

<a NAME="E2:13"/><b>then </b><i><font color="Green">E11</font></i>: 
( <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>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="tsep_2.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E3:13"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a> implies <font color="Maroon">c<sub>3</sub></font> 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/>

<b>assume </b><a NAME="E4:13"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>hence </b><a NAME="E5:13"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="tsep_2.html#E2:13"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


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