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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="E1:44"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V1">filled</a>
 <b>and </b><br/><a NAME="E2:44"/><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="E3:44"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E4:44"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 ;<br/>

<b>assume </b><a NAME="E5:44"/>
( not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E6:44"/><b>then </b>
( <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E7:44"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E8:44"/><i><font color="Green">E57</font></i>: 
 <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E9:44"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E10:44"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>,<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:44"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="fintopo4.html#T6">FINTOPO4:6</a>, </i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</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">C</font></span>)</span> = 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</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#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E13:44"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:44"><i><font color="Green">E27</font></i></a>, , <a class="ref" href="fintopo6.html#T1" target="_self">Th1</a>, , , </i>;<br/>


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