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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/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/>



<a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp;  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<div><i><font color="Green" title="E17">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">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><a NAME="E2:18_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">Q</font>
 <b>and </b><br/><a NAME="E3:18_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>and </b><br/><a NAME="E4:18_1"/><i><font color="Green" title="E20">A5</font></i>: 
( <font color="Maroon" title="c3">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="c4">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> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c5">P1</font> = <font color="Maroon" title="c3">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="connsp_1.html#E2:18_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">Q1</font> = <font color="Maroon" title="c4">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="connsp_1.html#E2:18_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E7:18_1"/>
<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</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#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="connsp_1.html#E2:18_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="connsp_1.html#E3:18_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="connsp_1.html#T7" target="_self" title="CONNSP_1:th.7">Th7</a></i>;<br/><a NAME="E8:18_1"/><b>then </b>
not <font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</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#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="connsp_1.html#E2:18_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="connsp_1.html#E4:18_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="connsp_1.html#D2" target="_self" title="CONNSP_1:def.2">Def2</a></i>;<br/><b>hence </b><a NAME="E9:18_1"/>
not <font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D3" target="_self" title="CONNSP_1:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_2"/>
not <font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ;<br/><a NAME="E2:18_2"/><b>then </b>
not <font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D3" target="_self" title="CONNSP_1:def.3">Def3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c3">P</font>, <font color="Maroon" title="c4">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E4:18_2"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">Q</font>
 <b>and </b><br/><a NAME="E5:18_2"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c4">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 <b>and </b><br/><a NAME="E6:18_2"/><i><font color="Green" title="E20">A8</font></i>: 
( <font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> &amp; <font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> )
 <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="c5">P1</font> = <font color="Maroon" title="c3">P</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">Q1</font> = <font color="Maroon" title="c4">Q</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> <b>by </b><i><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></i>;<br/><a NAME="E9:18_2"/>
<font color="Maroon" title="c5">P1</font>,<font color="Maroon" title="c6">Q1</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#E5:18_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="connsp_1.html#T6" target="_self" title="CONNSP_1:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E10:18_2"/>
 ex <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">Q1</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/>( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Q1</font> &amp; <font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">Q1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; <font color="Olive" title="b1">P1</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">Q1</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="txt" href="connsp_1.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="connsp_1.html#E4:18_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="connsp_1.html#E6:18_2"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
( <font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> iff for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Olive" title="b1">P</font>,<font color="Olive" title="b2">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  &amp; not <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">GX</font> holds <br/><font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="txt" href="connsp_1.html#E2:18"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>


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