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





<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green">E22</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/>

<a NAME="E3:5"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:5"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T25">PRE_TOPC:25</a></i>;<br/>
<a NAME="E5:5"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E31</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:5_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><a NAME="E3:5_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><b>hence </b><a NAME="E4:5_1"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:5_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><a NAME="E3:5_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><b>hence </b><a NAME="E4:5_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:5"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i>, , <a class="ref" href="pre_topc.html#T25">PRE_TOPC:25</a></i>;<br/>
<b>hence </b><a NAME="E9:5"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, , , , , <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a>, <a class="ref" href="pre_topc.html#T53">PRE_TOPC:53</a></i>;<br/>


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