<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">filled</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>;<br/>



<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V2" title="FIN_TOPO:attr.2">symmetric</a>
 ;<br/>

<b>thus </b><a NAME="E2:9"/>
( <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  implies <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> )
 <b>by </b><i><a class="ref" href="fintopo4.html#D1" target="_self" title="FINTOPO4:def.1">Def1</a></i>;<br/>

<b>thus </b><a NAME="E3:9"/>
( <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font>
 ;<br/>

<a NAME="E2:9_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:9"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="fintopo4.html#T8" target="_self" title="FINTOPO4:th.8">Th8</a></i>;<br/>
<b>hence </b><a NAME="E3:9_1"/>
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo4.html#E1:9_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="fintopo4.html#D1" target="_self" title="FINTOPO4:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>


</div>
