<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">X</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">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>set </b><font color="Maroon">B1</font> = <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font>;<br/>
<b>set </b><font color="Maroon">B2</font> = <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font>;<br/>
<a NAME="E1:73"/><i><font color="Green">E17</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<b>thus </b><a NAME="E2:73"/>
( <font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a>  implies  ex <font color="Olive">C1</font>, <font color="Olive">C2</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Olive">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">C1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">C2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">C</font> is <a href="pre_topc.html#V4">closed</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 ;<br/>

<a NAME="E2:73_1"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font>,<font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="tsep_1.html#T5" target="_self">Th5</a></i>;<br/>
<b>then consider </b><font color="Maroon">C1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E4:73_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> )
 <b>and </b><br/><a NAME="E5:73_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E6:73_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="txt" href="tsep_1.html#E8"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">C1</font>
;<br/>

<b>take </b>
<font color="Maroon">C2</font>
;<br/>

<b>take </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K3">`</a> ;<br/>

<a NAME="E7:73_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#D1" target="_self">Def1</a>, <a class="ref" href="tops_1.html#T37">TOPS_1:37</a></i>;<br/>
<a NAME="E8:73_1"/><i><font color="Green">E32</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E9:73_1"/>
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E10:73_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T50">XBOOLE_1:50</a></i>;<br/>
<a NAME="E11:73_1"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E12:73_1"/>
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E13:73_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>;<br/>
<a NAME="E14:73_1"/>
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T13">XBOOLE_1:13</a></i>;<br/>
<a NAME="E15:73_1"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T31">SUBSET_1:31</a></i>;<br/>
<a NAME="E16:73_1"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K5">\+\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D6">XBOOLE_0:def 6</a></i>;<br/>
<a NAME="E17:73_1"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T101">XBOOLE_1:101</a></i>;<br/>
<a NAME="E18:73_1"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T33">SUBSET_1:33</a></i>;<br/>
<a NAME="E19:73_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/>
<a NAME="E20:73_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E21:73_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T15" target="_self">Th15</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E22:73_1"/><b>then </b>
( <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>hence </b><a NAME="E23:73_1"/>
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D1" target="_self">Def1</a>, <a class="txt" href="tsep_1.html#E21"><i><font color="Green">Lemma55</font></i></a>, , <a class="ref" href="tsep_1.html#T16" target="_self">Th16</a>, <a class="ref" href="tops_1.html#T30">TOPS_1:30</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>


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

<b>given </b><font color="Maroon">C1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>, <font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E3:73"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> )
 <b>and </b><br/><a NAME="E4:73"/><i><font color="Green">E66</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:73"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C</font> is <a href="pre_topc.html#V4">closed</a> )
 ;<br/>

<a NAME="E6:73"/>
 ex <font color="Olive">C1</font>, <font color="Olive">C2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C1</font> &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C2</font> &amp; <font color="Olive">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">C2</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> &amp; <font color="Olive">C1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">C2</font> is <a href="pre_topc.html#V3">open</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">C1</font>
;<br/>

<b>take </b>
<font color="Maroon">C2</font>
;<br/>

<a NAME="E1:73_2"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E2:73_2"/><b>then </b>
( <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E3:73_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E4:73_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A2</font> &amp; <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>, <a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>;<br/>
<a NAME="E5:73_2"/><b>then </b>
( <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T35">XBOOLE_1:35</a></i>;<br/>
<a NAME="E6:73_2"/><b>then </b><i><font color="Green">E95</font></i>: 
( <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font></span>)</span> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T47">XBOOLE_1:47</a></i>;<br/>
<a NAME="E7:73_2"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> )
 
;<br/>
<a NAME="E8:73_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/>
<a NAME="E9:73_2"/><b>then </b><i><font color="Green">E96</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C1</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T43">XBOOLE_1:43</a></i>;<br/>
<a NAME="E10:73_2"/>
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E11:73_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E12:73_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E13:73_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E14:73_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<a NAME="E15:73_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T49">XBOOLE_1:49</a></i>;<br/>
<a NAME="E16:73_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T55">XBOOLE_1:55</a></i>;<br/>
<b>hence </b><a NAME="E17:73_2"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> &amp; <font color="Maroon">C1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">C2</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tsep_1.html#D1" target="_self">Def1</a>, <a class="txt" href="tsep_1.html#E21"><i><font color="Green">Lemma55</font></i></a>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:73"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A2</font>,<font color="Maroon">A2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:73"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="tsep_1.html#R2" target="_self">are_weakly_separated</a> 
 <b>by </b><i><a class="ref" href="tsep_1.html#T5" target="_self">Th5</a></i>;<br/>


</div>
