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<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/><b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">FT</font> is <a href="fin_topo.html#V1">filled</a> &amp; <font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a> )
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">FT</font> is <a href="fintopo6.html#V1" target="_self">connected</a>
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a>
 <b>and </b><br/><a NAME="E4:48"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">FT</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E5:48"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 ;<br/>

<a NAME="E6:48"/><i><font color="Green">E57</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">FT</font> is <a href="fin_topo.html#V5">connected</a>
 
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</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">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:48_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E2:48_1"/>
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E3:48_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="fintopo4.html#R1">are_separated</a> 
 ;<br/><i><font color="Green">E81</font></i>:  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">FT</font> = 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E3:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/><a NAME="E5:48_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><div><i><font color="Green">E83</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">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:48_1_2"/><b>then </b>
<font color="Maroon">A</font>,<font color="Maroon">Q</font> <a href="fintopo4.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E3:48_1_2"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">Q</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="fintopo4.html#T6">FINTOPO4:6</a></i>;<br/><a NAME="E4:48_1_2"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:48_1_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><a NAME="E6:48_1_2"/><b>then </b>
<font color="Maroon">Q</font>,<font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:48"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E7:48_1_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">Q</font>,<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E8:48_1_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">Q</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="fintopo4.html#T6">FINTOPO4:6</a></i>;<br/><a NAME="E9:48_1_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">FT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="fintopo6.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E10:48_1_2"/><b>then </b>
( <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> or <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="fintopo6.html#T6" target="_self">Th6</a>, </i>;<br/><b>hence </b><a NAME="E11:48_1_2"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> or <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T15">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">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:48_1_3"/><b>then </b>
<font color="Maroon">P</font>,<font color="Maroon">A</font> <a href="fintopo4.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E3:48_1_3"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="fintopo4.html#T6">FINTOPO4:6</a></i>;<br/><a NAME="E4:48_1_3"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:48"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:48_1_3"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><a NAME="E6:48_1_3"/><b>then </b>
<font color="Maroon">P</font>,<font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:48"><i><font color="Green">E31</font></i></a>, </i>;<br/><a NAME="E7:48_1_3"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E8:48_1_3"/><b>then </b><i><font color="Green">E98</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:48"><i><font color="Green">E27</font></i></a>, <a class="ref" href="fintopo4.html#T6">FINTOPO4:6</a></i>;<br/><a NAME="E9:48_1_3"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">FT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="fintopo6.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E10:48_1_3"/><b>then </b>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> or <font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="fintopo6.html#T6" target="_self">Th6</a>, </i>;<br/><b>hence </b><a NAME="E11:48_1_3"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> or <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:48_1"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> or <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i>, <a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, <a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="fintopo6.html#T4" target="_self">Th4</a>, , <a class="txt" href="fintopo6.html#E1:48"><i><font color="Green">E27</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:48"/>
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> is <a href="fin_topo.html#V5">connected</a>
 <b>by </b><i>, <a class="txt" href="fintopo6.html#E1:48"><i><font color="Green">E27</font></i></a></i>;<br/>


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