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

<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>thus </b><a NAME="E1:58"/>
( <font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a>  implies  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="tarski.html#R1">c=</a> <font color="Olive">C1</font> &amp; <font color="Maroon">A2</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> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</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> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">A1</font>,<font color="Maroon">A2</font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<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="E3:58_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> )
 <b>and </b><br/><a NAME="E4:58_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> )
 <b>and </b><br/><a NAME="E5:58_1"/><i><font color="Green">E29</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/>

<a NAME="E6:58_1"/>
( <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">A2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E7:58_1"/><b>then </b>
( <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> <font color="Maroon">A1</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> <font color="Maroon">A2</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<b>hence </b><a NAME="E8:58_1"/>
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">A2</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> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</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> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</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><b> such that </b><a NAME="E2:58"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> )
 <b>and </b><br/><a NAME="E3:58"/><i><font color="Green">E31</font></i>: 
<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> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font>
 <b>and </b><br/><a NAME="E4:58"/><i><font color="Green">E32</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> )
 ;<br/>

<a NAME="E5:58"/>
 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="tarski.html#R1">c=</a> <font color="Olive">C1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">C2</font> &amp; <font color="Olive">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp; <font color="Olive">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_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/>

<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_2_1"/>
<font color="Maroon">C1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A2</font>
 ;<br/><a NAME="E2:58_2_1"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="E3:58_2_1"/>
( <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">C2</font> &amp; <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">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="tsep_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E4:58_2_1"/><b>then </b>
( <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <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> <font color="Maroon">A2</font> &amp; <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> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <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"><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#T19">XBOOLE_1:19</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:58_2_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <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"><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#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:58_2_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">C2</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E7:58_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_2_2"/>
<font color="Maroon">C2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A1</font>
 ;<br/><a NAME="E2:58_2_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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="E3:58_2_2"/>
( <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</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> <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="tsep_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E4:58_2_2"/><b>then </b>
( <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <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> <font color="Maroon">A1</font> &amp; <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> <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <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"><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#T19">XBOOLE_1:19</a>, <a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:58_2_2"/><b>then </b>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <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"><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#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E6:58_2_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"><font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A2</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E7:58_2_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:58_2"/>
( <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C1</font> &amp; <font color="Maroon">A2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">C1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A2</font> &amp; <font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</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> )
 <b>by </b><i><a class="ref" href="tsep_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="tsep_1.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:58"/>
<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#E8"><i><font color="Green">Lemma35</font></i></a></i>;<br/>


</div>
