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

<b>let </b><font color="Maroon" title="c1">FT</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>  st <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> holds <br/>( <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  iff ( <font color="Olive" title="b1">A</font> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">B</font> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a>  ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> 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>  iff ( <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> &amp; <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#K12" title="FIN_TOPO:func.12">^f</a>  ) ) )</span><br/>



<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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>  iff ( <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> &amp; <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#K12" title="FIN_TOPO:func.12">^f</a>  ) )</span><br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green" title="E5">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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T17" title="FIN_TOPO:th.17">FIN_TOPO:17</a></i>;<br/>
<a NAME="E3:7"/>
<font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a> 
 
<b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="fin_topo.html#T17" title="FIN_TOPO:th.17">FIN_TOPO:17</a></i>;<br/>
<b>hence </b><a NAME="E4:7"/>
( <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>  iff ( <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K12" title="FIN_TOPO:func.12">^f</a>  <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">B</font> &amp; <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#K12" title="FIN_TOPO:func.12">^f</a>  ) )
 <b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="fintopo4.html#D1" target="_self" title="FINTOPO4:def.1">Def1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
