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<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="15_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">A1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><a NAME="E1:15_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">A1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A1</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="Maroon">A1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> )
 ;<br/><a NAME="E2:15_1"/><i><font color="Green">E25</font></i>: 
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</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">A1</font></span>)</span> <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">A1</font> )
 <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/><a NAME="E3:15_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A1</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/><a NAME="E4:15_1"/>
<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#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">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_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E5:15_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A1</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">A1</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E6:15_1"/>
not <font color="Maroon">GX</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_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">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:15_2"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E4:15_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>and </b><br/><a NAME="E5:15_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">P</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="Maroon">Q</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/>;<br/><b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font>, <font color="Maroon">Q</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> ;<br/><a NAME="E7:15_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">P</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, , </i>;<br/><a NAME="E8:15_2"/>
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>by </b><i>, </i>;<br/><a NAME="E9:15_2"/><b>then </b>
<font color="Maroon">Q</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">P</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T25">PRE_TOPC:25</a></i>;<br/><a NAME="E10:15_2"/><b>then </b>
<font color="Maroon">P</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="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/><b>hence </b><a NAME="E11:15_2"/>
 ex <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">P</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">P</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">P</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">P</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>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
( <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> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; not <font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font> holds <br/><font color="Olive">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">GX</font> )
 <b>by </b><i/>;<br/>


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