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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="E1:48"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<a NAME="E2:48"/>
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:48"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="txt" href="tsep_1.html#E4:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="tops_1.html#T31">TOPS_1:31</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green">E27</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/>
<b>assume </b><a NAME="E5:48"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/>

<a NAME="E6:48"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#D1">CONNSP_1:def 1</a></i>;<br/>
<a NAME="E7:48"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a>, <a class="ref" href="xboole_1.html#T83">XBOOLE_1:83</a></i>;<br/>
<b>hence </b><a NAME="E8:48"/>
( <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> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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