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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:12_1"/><i><font color="Green">E14</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E2:12_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>and </b><br/><a NAME="E3:12_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/><a NAME="E4:12_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:12_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="connsp_1.html#E3:12_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a></i>;<br/><b>hence </b><a NAME="E5:12_1"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:12_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="connsp_1.html#E2:12_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/>
not <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green">E14</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E4:12_2"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>and </b><br/><a NAME="E5:12_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E7:12_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:12_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="connsp_1.html#E4:12_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a></i>;<br/><a NAME="E8:12_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E4:12_2"><i><font color="Green">E15</font></i></a>, <a class="ref" href="connsp_1.html#T2" target="_self">Th2</a></i>;<br/><b>hence </b><a NAME="E9:12_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:12_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="connsp_1.html#E5:12_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="connsp_1.html#E7:12_2"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:12"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a> iff for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:12"><i><font color="Green">E13</font></i></a></i>;<br/>


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