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<b>let </b><font color="Maroon">X</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">A2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<b>assume </b><a NAME="E1:47"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">A1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A2</font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<b>thus </b><a NAME="E2:47"/>
( <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> implies <font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font>
 ;<br/>

<a NAME="E2:47_1"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a> 
 <b>by </b><i><a class="txt" href="tsep_1.html#E1:47_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#D1">CONNSP_1:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:47"/>
( <font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a>  implies <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> )
 <b>by </b><i><a class="ref" href="connsp_1.html#T2">CONNSP_1:2</a></i>;<br/>


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