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<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="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><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">filled</a> &amp;  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">FT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> )
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E21">A2</font></i>: 
<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> 
 ;<br/>

<a NAME="E3:39"/><i><font color="Green" title="E22">A3</font></i>: 
<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>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E2:39"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="fintopo4.html#T6" title="FINTOPO4:th.6">FINTOPO4:6</a>, <a class="txt" href="fintopo6.html#E1:39"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="fin_topo.html#T18" title="FIN_TOPO:th.18">FIN_TOPO:18</a>, <a class="txt" href="fintopo6.html#E1:39"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>
<div><i><font color="Green" title="E24">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/>
<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="E2:39_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:39"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/><b>hence </b><a NAME="E3:39_1"/>
<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>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:39"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E25">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
<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> 
 ;<br/><a NAME="E2:39_2"/><b>then </b>
<font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">B</font>
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:39"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xboole_1.html#T73" title="XBOOLE_1:th.73">XBOOLE_1:73</a></i>;<br/><b>hence </b><a NAME="E3:39_2"/>
<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>
 <b>by </b><i><a class="txt" href="fintopo6.html#E4:39"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:39"/><i><font color="Green" title="E26">A7</font></i>: 
( <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">closed</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">closed</a> )
 
<b>by </b><i><a class="ref" href="fin_topo.html#D17" title="FIN_TOPO:def.17">FIN_TOPO:def 17</a>, <a class="txt" href="fintopo6.html#E5:39"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="fintopo6.html#E6:39"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="fintopo6.html#E2:39"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="fintopo4.html#D1" title="FINTOPO4:def.1">FINTOPO4:def 1</a></i>;<br/>
<a NAME="E8:39"/>
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E1:39"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="fintopo6.html#E3:39"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="pre_topc.html#T25" title="PRE_TOPC:th.25">PRE_TOPC:25</a></i>;<br/>
<b>hence </b><a NAME="E9:39"/>
( <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">open</a> &amp; <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E7:39"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="fin_topo.html#T31" title="FIN_TOPO:th.31">FIN_TOPO:31</a></i>;<br/>


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