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<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</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">FT</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E27">A1</font></i>: 
( <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">filled</a> &amp; <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V2" title="FIN_TOPO:attr.2">symmetric</a> )
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c1">FT</font> is <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V5" title="FIN_TOPO:attr.5">connected</a>
 <b>and </b><br/><a NAME="E4:48"/><i><font color="Green" title="E30">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">FT</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="E5:48"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c3">B</font>,<font color="Maroon" title="c4">C</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 ;<br/>

<a NAME="E6:48"/><i><font color="Green" title="E32">A6</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V5" title="FIN_TOPO:attr.5">connected</a>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E2:48"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="fintopo6.html#D1" target="_self" title="FINTOPO6:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">P</font>, <font color="Maroon" title="c6">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:48_1"/><i><font color="Green" title="E33">A7</font></i>: 
<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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Q</font>
 <b>and </b><br/><a NAME="E2:48_1"/>
<font color="Maroon" title="c5">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q</font>
 <b>and </b><br/><a NAME="E3:48_1"/><i><font color="Green" title="E34">A8</font></i>: 
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 ;<br/><i><font color="Green" title="E35">A9</font></i>:  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">FT</font> = 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><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></span>)</span>
<b>by </b><i><a class="txt" href="fintopo6.html#E4:48"><i><font color="Green" title="E30">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="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Q</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E1:48_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>
;<br/><a NAME="E5:48_1"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Q</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><div><i><font color="Green" title="E37">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_2"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">P</font>
 ;<br/><a NAME="E2:48_1_2"/><b>then </b>
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c6">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="fintopo6.html#T29" target="_self" title="FINTOPO6:th.29">Th29</a></i>;<br/><a NAME="E3:48_1_2"/><b>then </b><i><font color="Green" title="E38">A12</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="fintopo4.html#T6" title="FINTOPO4:th.6">FINTOPO4:6</a></i>;<br/><a NAME="E4:48_1_2"/>
<font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:48_1_2"/><b>then </b>
<font color="Maroon" title="c6">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48_1_2"><i><font color="Green" title="E38">A12</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/><a NAME="E6:48_1_2"/><b>then </b>
<font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c4">C</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E5:48"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="fintopo6.html#T29" target="_self" title="FINTOPO6:th.29">Th29</a></i>;<br/><a NAME="E7:48_1_2"/><b>then </b><i><font color="Green" title="E39">A13</font></i>: 
<font color="Maroon" title="c6">Q</font>,<font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="fintopo6.html#T30" target="_self" title="FINTOPO6:th.30">Th30</a></i>;<br/><a NAME="E8:48_1_2"/><b>then </b><i><font color="Green" title="E40">A14</font></i>: 
<font color="Maroon" title="c6">Q</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="fintopo4.html#T6" title="FINTOPO4:th.6">FINTOPO4:6</a></i>;<br/><a NAME="E9:48_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">FT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:48_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/><a NAME="E10:48_1_2"/><b>then </b>
( <font color="Maroon" title="c6">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">FT</font> or <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">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">FT</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:48"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="fintopo6.html#E7:48_1_2"><i><font color="Green" title="E39">A13</font></i></a>, <a class="txt" href="fintopo6.html#E8:48_1_2"><i><font color="Green" title="E40">A14</font></i></a>, <a class="ref" href="fintopo6.html#T4" target="_self" title="FINTOPO6:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E11:48_1_2"/>
( <font color="Maroon" title="c5">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">FT</font> or <font color="Maroon" title="c6">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">FT</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="48_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1_3"/>
<font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">Q</font>
 ;<br/><a NAME="E2:48_1_3"/><b>then </b>
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">A</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="fintopo6.html#T29" target="_self" title="FINTOPO6:th.29">Th29</a></i>;<br/><a NAME="E3:48_1_3"/><b>then </b><i><font color="Green" title="E38">A15</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="fintopo4.html#T6" title="FINTOPO4:th.6">FINTOPO4:6</a></i>;<br/><a NAME="E4:48_1_3"/>
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:48_1_3"/><b>then </b>
<font color="Maroon" title="c5">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48_1_3"><i><font color="Green" title="E38">A15</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/><a NAME="E6:48_1_3"/><b>then </b>
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c4">C</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E5:48"><i><font color="Green" title="E31">A5</font></i></a>, <a class="ref" href="fintopo6.html#T29" target="_self" title="FINTOPO6:th.29">Th29</a></i>;<br/><a NAME="E7:48_1_3"/><b>then </b><i><font color="Green" title="E39">A16</font></i>: 
<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="ref" href="fintopo6.html#T30" target="_self" title="FINTOPO6:th.30">Th30</a></i>;<br/><a NAME="E8:48_1_3"/><b>then </b><i><font color="Green" title="E40">A17</font></i>: 
<font color="Maroon" title="c5">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="fintopo4.html#T6" title="FINTOPO4:th.6">FINTOPO4:6</a></i>;<br/><a NAME="E9:48_1_3"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">FT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:48_1"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="xboole_1.html#T4" title="XBOOLE_1:th.4">XBOOLE_1:4</a></i>;<br/><a NAME="E10:48_1_3"/><b>then </b>
( <font color="Maroon" title="c5">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">FT</font> or <font color="Maroon" title="c6">Q</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">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">FT</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E6:48"><i><font color="Green" title="E32">A6</font></i></a>, <a class="txt" href="fintopo6.html#E7:48_1_3"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="fintopo6.html#E8:48_1_3"><i><font color="Green" title="E40">A17</font></i></a>, <a class="ref" href="fintopo6.html#T4" target="_self" title="FINTOPO6:th.4">Th4</a></i>;<br/><b>hence </b><a NAME="E11:48_1_3"/>
( <font color="Maroon" title="c5">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">FT</font> or <font color="Maroon" title="c6">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">FT</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="E8:48_1"/>
( <font color="Maroon" title="c5">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">FT</font> or <font color="Maroon" title="c6">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">FT</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="fintopo6.html#E3:48_1"><i><font color="Green" title="E34">A8</font></i></a>, <a class="txt" href="fintopo6.html#E5:48_1"><i><font color="Green" title="E36">A10</font></i></a>, <a class="txt" href="fintopo6.html#E6:48_1"><i><font color="Green" title="E37">A11</font></i></a>, <a class="ref" href="fintopo6.html#T33" target="_self" title="FINTOPO6:th.33">Th33</a>, <a class="txt" href="fintopo6.html#E1:48"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:48"/>
<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="fin_topo.html#V5" title="FIN_TOPO:attr.5">connected</a>
 <b>by </b><i><a class="ref" href="fintopo6.html#T7" target="_self" title="FINTOPO6:th.7">Th7</a>, <a class="txt" href="fintopo6.html#E1:48"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>


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