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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>, <font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/>







<div><i><font color="Green" title="E21">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">B</font>, <font color="Maroon" title="c7">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:23_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>and </b><br/><a NAME="E2:23_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c5">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E3:23_1"/><i><font color="Green" title="E24">A4</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font>
 <b>and </b><br/><a NAME="E4:23_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c7">C</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">P</font>, <font color="Maroon" title="c9">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:23_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q</font>
 <b>and </b><br/><a NAME="E2:23_1_1"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ;<br/><i><font color="Green" title="E28">A8</font></i>:  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font> = 
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="xboole_1.html#T45" title="XBOOLE_1:th.45">XBOOLE_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c9">Q</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font>
<b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
;<br/><div><i><font color="Green" title="E29">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_2"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">P</font>
 ;<br/><a NAME="E2:23_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">Q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="connsp_1.html#T2" target="_self" title="CONNSP_1:th.2">Th2</a>, <a class="ref" href="connsp_1.html#T8" target="_self" title="CONNSP_1:th.8">Th8</a></i>;<br/><a NAME="E3:23_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/><a NAME="E4:23_1_1_2"/><b>then </b>
<font color="Maroon" title="c9">Q</font>,<font color="Maroon" title="c7">C</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#E4:23_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self" title="CONNSP_1:th.8">Th8</a></i>;<br/><a NAME="E5:23_1_1_2"/><b>then </b><i><font color="Green" title="E30">A10</font></i>: 
<font color="Maroon" title="c9">Q</font>,<font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</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#E2:23_1_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="connsp_1.html#T9" target="_self" title="CONNSP_1:th.9">Th9</a></i>;<br/><a NAME="E6:23_1_1_2"/>
 <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="c9">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/><a NAME="E7:23_1_1_2"/><b>then </b>
( <font color="Maroon" title="c9">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> or <font color="Maroon" title="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</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#E1:23_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="connsp_1.html#E5:23_1_1_2"><i><font color="Green" title="E30">A10</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="E8:23_1_1_2"/>
( <font color="Maroon" title="c8">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> or <font color="Maroon" title="c9">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="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_3"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">Q</font>
 ;<br/><a NAME="E2:23_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="connsp_1.html#T2" target="_self" title="CONNSP_1:th.2">Th2</a>, <a class="ref" href="connsp_1.html#T8" target="_self" title="CONNSP_1:th.8">Th8</a></i>;<br/><a NAME="E3:23_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">B</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/><a NAME="E4:23_1_1_3"/><b>then </b>
<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c7">C</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#E4:23_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self" title="CONNSP_1:th.8">Th8</a></i>;<br/><a NAME="E5:23_1_1_3"/><b>then </b><i><font color="Green" title="E30">A11</font></i>: 
<font color="Maroon" title="c8">P</font>,<font color="Maroon" title="c9">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</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#E2:23_1_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="connsp_1.html#T9" target="_self" title="CONNSP_1:th.9">Th9</a></i>;<br/><a NAME="E6:23_1_1_3"/>
 <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="c8">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/><a NAME="E7:23_1_1_3"/><b>then </b>
( <font color="Maroon" title="c8">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> or <font color="Maroon" title="c9">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">C</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#E1:23_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="connsp_1.html#E5:23_1_1_3"><i><font color="Green" title="E30">A11</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="E8:23_1_1_3"/>
( <font color="Maroon" title="c8">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> or <font color="Maroon" title="c9">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="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:23_1_1"/>
( <font color="Maroon" title="c8">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> or <font color="Maroon" title="c9">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#E2:23_1"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="connsp_1.html#E4:23_1_1"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="connsp_1.html#T17" target="_self" title="CONNSP_1:th.17">Th17</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:23_1"/>
<font color="Maroon" title="c5">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">B</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#T16" target="_self" title="CONNSP_1:th.16">Th16</a></i>;<br/></div><b>end;</b></div>
<b>assume </b><b>that </b><br/><a NAME="E2:23"/><i><font color="Green" title="E22">A12</font></i>: 
<font color="Maroon" title="c1">GX</font> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green" title="E23">A13</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green" title="E24">A14</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">GX</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green" title="E25">A15</font></i>: 
<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ;<br/>

<b>thus </b><a NAME="E6:23"/>
( <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> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="connsp_1.html#E2:23"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="connsp_1.html#E3:23"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="connsp_1.html#E4:23"><i><font color="Green" title="E24">A14</font></i></a>, <a class="txt" href="connsp_1.html#E5:23"><i><font color="Green" title="E25">A15</font></i></a></i>;<br/>


</div>
