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

<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><a NAME="E1:14_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E2:14_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E3:14_1"/><i><font color="Green">E27</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 ;<br/><a NAME="E4:14_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<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></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_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:14_1"/><i><font color="Green">E30</font></i>: 
<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> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/><a NAME="E6:14_1"/>
( <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> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <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> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E7:14_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E8:14_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<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></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#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E9:14_1"/><b>then </b><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> <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="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E10:14_1"/>
( <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;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E11:14_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E12:14_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><span class="p3">(<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></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E13:14_1"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E14:14_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font>,<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> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="ref" href="connsp_1.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E15:14_1"/>
 <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> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/><a NAME="E16:14_1"/><b>then </b>
 <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> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span>
 ;<br/><b>hence </b><a NAME="E17:14_1"/>
not <font color="Maroon">GX</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i>, , <a class="ref" href="connsp_1.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:14_2"/>
not <font color="Maroon">GX</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:14_2"/><i><font color="Green">E33</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="E4:14_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E5:14_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E6:14_2"/><i><font color="Green">E48</font></i>: 
( <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#V4">closed</a> )
 <b>and </b><br/><a NAME="E7:14_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i/>;<br/><a NAME="E8:14_2"/><i><font color="Green">E53</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="E9:14_2"/><i><font color="Green">E54</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <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="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E10:14_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D1" target="_self">Def1</a>, <a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/><b>hence </b><a NAME="E11:14_2"/>
 ex <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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="Olive">A</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="connsp_1.html#D1" target="_self">Def1</a>, <a class="ref" href="connsp_1.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:14"/>
( <font color="Maroon">GX</font> is <a href="connsp_1.html#V1" target="_self">connected</a> iff for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> holds <br/> <a href="pre_topc.html#K4">Cl</a> <font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><span class="p2">(<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="Olive">A</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div>
