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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>assume </b><a NAME="E1:10"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a>
 ;<br/>

<b>assume </b><a NAME="E2:10"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">A2</font>, <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">A2</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B2</font> &amp; <font color="Olive">A2</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B2</font> &amp; <font color="Olive">A2</font>,<font color="Olive">B2</font> <a href="fintopo4.html#R1">are_separated</a>  &amp; not <font color="Olive">A2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> holds <br/><font color="Olive">B2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font>
 ;<br/>

<a NAME="E3:10"/>
for <font color="Olive">B</font>, <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">C</font> &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">C</font> holds <br/>( <font color="Olive">B</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">C</font> &amp; <font color="Olive">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">C</font> <a href="fin_topo.html#K9">^b</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:10_1"/>
( <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> &amp; <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">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:10_1"/><b>then </b>
not <font color="Maroon">B</font>,<font color="Maroon">C</font> <a href="fintopo4.html#R1">are_separated</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:10_1"/><b>then </b>
(  not <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> or  not <font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> <a href="fin_topo.html#K9">^b</a>  )
 
<b>by </b><i><a class="ref" href="fintopo4.html#D1">FINTOPO4:def 1</a></i>;<br/>
<b>hence </b><a NAME="E4:10_1"/>
( <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> <a href="fin_topo.html#K9">^b</a>  )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:10"/>
<font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a>
 <b>by </b><i/>;<br/>


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