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<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a> <a href="fin_topo.html#V1">filled</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>



<b>assume </b><a NAME="E1:26"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/>
not <font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a>
 ;<br/><a NAME="E2:26_1"/><b>then </b>
not <font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font> is <a href="fintopo6.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/><a NAME="E3:26_1"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> is <a href="fin_topo.html#V5">connected</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E5:26_1"/><i><font color="Green">E31</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E6:26_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E7:26_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E8:26_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/><b>reconsider </b><font color="Maroon">P1</font> = <font color="Maroon">P</font>, <font color="Maroon">Q1</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font> <b>by </b><i/>;<br/><b>take </b><font color="Maroon">P1</font> = <font color="Maroon">P1</font>;<br/><b>take </b><font color="Maroon">Q1</font> = <font color="Maroon">Q1</font>;<br/><b>thus </b><a NAME="E10:26_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q1</font> &amp; <font color="Maroon">P1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E11:26_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/><a NAME="E12:26_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">P</font> <a href="fin_topo.html#K9">^b</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P1</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
<br/>.= 
 <a href="xboole_0.html#K1">{}</a> 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>
;<br/><a NAME="E14:26_1"/><b>then </b>
<font color="Maroon">P1</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E15:26_1"/>
( <font color="Maroon">P1</font>,<font color="Maroon">Q1</font> <a href="fintopo4.html#R1">are_separated</a>  &amp; <font color="Maroon">P1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> &amp; <font color="Maroon">Q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i>, <a class="ref" href="fintopo4.html#T10">FINTOPO4:10</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:26"/>
( <font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a> iff for <font color="Olive">P</font>, <font color="Olive">Q</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">P</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">Q</font> &amp; <font color="Olive">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">Q</font> &amp; <font color="Olive">P</font>,<font color="Olive">Q</font> <a href="fintopo4.html#R1">are_separated</a>  &amp; not <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> holds <br/><font color="Olive">Q</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">FT</font> )
 <b>by </b><i/>;<br/>


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