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<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"> ( <font color="Maroon" title="c1">FT</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> &amp; ( 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  <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="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Olive" title="b1">A</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> &amp; <font color="Olive" title="b2">B</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</font> ) implies <font color="Maroon" title="c1">FT</font> is <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a> )</span><br/>

<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c1">FT</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> &amp; <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"> (  ex <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 <br/>(  <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="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Olive" title="b1">A</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> &amp; <font color="Olive" title="b2">B</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> &amp; not <font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</font> ) or <font color="Maroon" title="c1">FT</font> is <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a> )</span><br/>

<b>assume </b><a NAME="E2:42"/><i><font color="Green" title="E23">A2</font></i>: 
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  <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="Olive" title="b1">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b1">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Olive" title="b2">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Olive" title="b1">A</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> &amp; <font color="Olive" title="b2">B</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> holds <br/><font color="Olive" title="b1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">B</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="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a></span><br/>

<b>assume </b><a NAME="E3:42"/>
not <font color="Maroon" title="c1">FT</font> is <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:42"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>
 
<b>by </b><i><a class="ref" href="fintopo6.html#D1" target="_self" title="FINTOPO6:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">P</font>, <font color="Maroon" title="c3">Q</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font><b> such that </b><br/><a NAME="E6:42"/><i><font color="Green" title="E24">A3</font></i>: 
 <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">P</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">Q</font>
 <b>and </b><br/><a NAME="E7:42"/><i><font color="Green" title="E25">A4</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">Q</font> <a href="fintopo4.html#R1" title="FINTOPO4:pred.1">are_separated</a>  )
 <b>and </b><br/><a NAME="E8:42"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c2">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Maroon" title="c3">Q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">FT</font> )
 <b>by </b><i><a class="txt" href="fintopo6.html#E1:42"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="fintopo6.html#T7" target="_self" title="FINTOPO6:th.7">Th7</a></i>;<br/>
<a NAME="E9:42"/>
( <font color="Maroon" title="c2">P</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> &amp; <font color="Maroon" title="c3">Q</font> is <a href="fin_topo.html#V3" title="FIN_TOPO:attr.3">closed</a> )
 
<b>by </b><i><a class="txt" href="fintopo6.html#E1:42"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="fintopo6.html#E6:42"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="fintopo6.html#E7:42"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="fintopo6.html#T28" target="_self" title="FINTOPO6:th.28">Th28</a></i>;<br/>
<b>hence </b><a NAME="E10:42"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:42"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="fintopo6.html#E6:42"><i><font color="Green" title="E24">A3</font></i></a>, <a class="txt" href="fintopo6.html#E7:42"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="fintopo6.html#E8:42"><i><font color="Green" title="E26">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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