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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>, <font color="Maroon">B</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="E1:45"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a>
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a>
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">B</font> is <a href="fin_topo.html#V5">connected</a>
 <b>and </b><br/><a NAME="E4:45"/><i><font color="Green">E41</font></i>: 
not <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="fintopo4.html#R1">are_separated</a> 
 ;<br/>

<b>given </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font><b> such that </b><a NAME="E5:45"/><i><font color="Green">E43</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="E6:45"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E7:45"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E8:45"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E9:45"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a><br/>

<a NAME="E10:45"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Q</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , </i>;<br/>
<div><i><font color="Green">E83</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 ;<br/><a NAME="E2:45_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1_1"/>
( <font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a>  )
 </span><b>by </b><i><a class="txt" href="fintopo6.html#E1:45"><i><font color="Green">E27</font></i></a>, <a class="ref" href="fintopo4.html#D1">FINTOPO4:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_1"><b>suppose </b></a><a NAME="E1:45_1_1_1"/>
<font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:45_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="fintopo6.html#T3" target="_self">Th3</a>, <a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1_2"><b>suppose </b></a><a NAME="E1:45_1_1_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a> 
 ;<br/><div class="add"><a NAME="E2:45_1_1_2"/><b>then </b>
not <font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:45_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="fintopo6.html#T3" target="_self">Th3</a>, <a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E2:45_2"/><b>then </b><i><font color="Green">E97</font></i>: 
<font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="fin_topo.html#K9">^b</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_2_1"/>
( <font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a>  )
 </span><b>by </b><i><a class="txt" href="fintopo6.html#E1:45"><i><font color="Green">E27</font></i></a>, <a class="ref" href="fintopo4.html#D1">FINTOPO4:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1_1"><b>suppose </b></a><a NAME="E1:45_2_1_1"/>
<font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:45_2_1_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><b>hence </b><a NAME="E3:45_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:45"><i><font color="Green">E33</font></i></a>, <a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_2_1_2"><b>suppose </b></a><a NAME="E1:45_2_1_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a> 
 ;<br/><div class="add"><a NAME="E2:45_2_1_2"/><b>then </b>
not <font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:45_2_1_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="fintopo6.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/><b>hence </b><a NAME="E4:45_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E3:45"><i><font color="Green">E33</font></i></a>, <a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E98</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_3"/><i><font color="Green">E99</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then </b><i><font color="Green">E100</font></i>: <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>

<br/>.= 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <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>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><i><font color="Green">E101</font></i>: <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="fintopo6.html#T4" target="_self">Th4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><div><i><font color="Green">E102</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_3_3"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then </b><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> = 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span>

<br/>.= 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><b>hence </b><a NAME="E3:45_3_3"/>
contradiction
 <b>by </b><i><a class="ref" href="fintopo6.html#T3" target="_self">Th3</a>, , <a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">P3</font> = <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>;<br/><b>set </b><font color="Maroon">Q3</font> = <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>;<br/><i><font color="Green">E105</font></i>: <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> = 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <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>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><a NAME="E6:45_3"/><i><font color="Green">E106</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T76">XBOOLE_1:76</a>, <a class="txt" href="fintopo6.html#E2:45"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E7:45_3"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E8:45_3"/><b>then </b><i><font color="Green">E107</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="fin_topo.html#K9">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/><a NAME="E9:45_3"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E10:45_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a>, <a class="txt" href="fintopo6.html#E5:45"><i><font color="Green">E43</font></i></a>, </i>;<br/><b>hence </b><a NAME="E11:45_3"/>
contradiction
 <b>by </b><i>, <a class="ref" href="fintopo6.html#T6" target="_self">Th6</a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, , , <a class="txt" href="fintopo6.html#E4:45"><i><font color="Green">E41</font></i></a>, <a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E108</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then </b><i><font color="Green">E110</font></i>: <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span>

<br/>.= 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><i><font color="Green">E111</font></i>: <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">Q</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><div><i><font color="Green">E112</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_4_3"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then </b><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> = 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span>

<br/>.= 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <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>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><b>hence </b><a NAME="E3:45_4_3"/>
contradiction
 <b>by </b><i><a class="ref" href="fintopo6.html#T2" target="_self">Th2</a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T18">XBOOLE_1:18</a></i>;<br/></div><b>end;</b></div><b>set </b><font color="Maroon">P3</font> = <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>;<br/><b>set </b><font color="Maroon">Q3</font> = <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>;<br/><i><font color="Green">E113</font></i>: <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> = 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <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>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/><a NAME="E6:45_4"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T76">XBOOLE_1:76</a>, <a class="txt" href="fintopo6.html#E2:45"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E7:45_4"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E8:45_4"/><b>then </b><i><font color="Green">E115</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <a href="fin_topo.html#K9">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="fin_topo.html#K9">^b</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/><a NAME="E9:45_4"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E10:45_4"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a>, , <a class="txt" href="fintopo6.html#E3:45"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E11:45_4"/>
contradiction
 <b>by </b><i>, <a class="txt" href="fintopo6.html#E4:45"><i><font color="Green">E41</font></i></a>, <a class="ref" href="fintopo6.html#T6" target="_self">Th6</a>, , <a class="txt" href="fintopo6.html#E5:45"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/></div><b>end;</b></div>
<b>set </b><font color="Maroon">P2</font> = <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font>;<br/>
<b>set </b><font color="Maroon">Q2</font> = <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>;<br/>
<i><font color="Green">E116</font></i>: <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font></span>)</span> = 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <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>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>
;<br/>
<a NAME="E16:45"/><i><font color="Green">E117</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T76">XBOOLE_1:76</a>, <a class="txt" href="fintopo6.html#E2:45"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E17:45"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E18:45"/><b>then </b><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="fin_topo.html#K9">^b</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T19">FIN_TOPO:19</a></i>;<br/>
<a NAME="E19:45"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E20:45"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">P</font></span>)</span> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T64">XBOOLE_1:64</a>, <a class="ref" href="fintopo6.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>hence </b><a NAME="E21:45"/>
contradiction
 <b>by </b><i><a class="ref" href="fintopo6.html#D1" target="_self">Def1</a>, <a class="ref" href="fintopo6.html#T4" target="_self">Th4</a>, <a class="ref" href="fintopo6.html#T5" target="_self">Th5</a>, , <a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a>, </i>;<br/>


</div>
