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<b>let </b><font color="Maroon" title="c1">GX</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<div><i><font color="Green" title="E13">A1</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" title="c2">A</font>, <font color="Maroon" title="c3">B</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><a NAME="E1:12_1"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E2:12_1"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> )
 <b>and </b><br/><a NAME="E3:12_1"/><i><font color="Green" title="E16">A4</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 ;<br/><a NAME="E4:12_1"/>
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:12_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="connsp_1.html#E3:12_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self" title="CONNSP_1:th.3">Th3</a></i>;<br/><b>hence </b><a NAME="E5:12_1"/>
not <font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:12_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="connsp_1.html#E2:12_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="connsp_1.html#D2" target="_self" title="CONNSP_1:def.2">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" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 ;<br/><b>then consider </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green" title="E14">A5</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font>
 <b>and </b><br/><a NAME="E4:12_2"/><i><font color="Green" title="E15">A6</font></i>: 
<font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>and </b><br/><a NAME="E5:12_2"/><i><font color="Green" title="E16">A7</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> )
 <b>by </b><i><a class="ref" href="connsp_1.html#D2" target="_self" title="CONNSP_1:def.2">Def2</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c4">P</font> = <font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c5">Q</font> = <font color="Maroon" title="c3">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> ;<br/><a NAME="E7:12_2"/><i><font color="Green" title="E17">A8</font></i>: 
( <font color="Maroon" title="c4">P</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c5">Q</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:12_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="connsp_1.html#E4:12_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self" title="CONNSP_1:th.5">Th5</a></i>;<br/><a NAME="E8:12_2"/>
<font color="Maroon" title="c4">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E4:12_2"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="connsp_1.html#T2" target="_self" title="CONNSP_1:th.2">Th2</a></i>;<br/><b>hence </b><a NAME="E9:12_2"/>
 ex <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> st <br/>(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:12_2"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="connsp_1.html#E5:12_2"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="connsp_1.html#E7:12_2"><i><font color="Green" title="E17">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:12"/>
( <font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> iff for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>


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