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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="fin_topo.html#V2">filled</a>  <a href="fin_topo.html#L1">FT_Space_Str</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:10"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="fin_topo.html#V3">symmetric</a>
 ;<br/>

<b>thus </b><a NAME="E2:10"/>
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="fintopo4.html#R1" target="_self">are_separated</a>  implies <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="fintopo4.html#D1" target="_self">Def1</a></i>;<br/>

<b>thus </b><a NAME="E3:10"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="fintopo4.html#R1" target="_self">are_separated</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="fin_topo.html#K8">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:10_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> <a href="fin_topo.html#K8">^b</a> 
 
<b>by </b><i><a class="txt" href="fintopo4.html#E1:10"><i><font color="Green">E6</font></i></a>, <a class="ref" href="fintopo4.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E3:10_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="fintopo4.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="fintopo4.html#E1:10_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="fintopo4.html#D1" target="_self">Def1</a></i>;<br/>


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


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