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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>assume </b><a NAME="E1:42"/><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> )
 ;<br/>

<b>assume </b><a NAME="E2:42"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">FT</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> &amp; <font color="Olive">A</font> is <a href="fin_topo.html#V4">closed</a> &amp; <font color="Olive">B</font> is <a href="fin_topo.html#V4">closed</a> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B</font>
 ;<br/>

<b>assume </b><a NAME="E3:42"/>
not <font color="Maroon">FT</font> is <a href="fintopo6.html#V1" target="_self">connected</a>
 ;<br/>

<a NAME="E4:42"/><b>then </b>
not  <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/>;<br/>
<b>then consider </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><br/><a NAME="E6:42"/><i><font color="Green">E33</font></i>: 
 <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> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E7:42"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="fintopo4.html#R1">are_separated</a>  )
 <b>and </b><br/><a NAME="E8:42"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:42"/>
( <font color="Maroon">P</font> is <a href="fin_topo.html#V4">closed</a> &amp; <font color="Maroon">Q</font> is <a href="fin_topo.html#V4">closed</a> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:42"/>
contradiction
 <b>by </b><i>, , , </i>;<br/>


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