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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"> 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> &amp; <font color="Olive" title="b1">A</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Olive" title="b2">B</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> &amp; not <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> 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> </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> &amp; <font color="Maroon" title="c2">A</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Maroon" title="c3">B</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> &amp; not <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> 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>  )</span><br/>



<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E29">A1</font></i>: 
( <font color="Maroon" title="c1">FT</font> is <a href="fin_topo.html#V1" title="FIN_TOPO:attr.1">symmetric</a> &amp; <font color="Maroon" title="c2">A</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> &amp; <font color="Maroon" title="c3">B</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <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="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> or <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>  )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:51"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E3:51"/><i><font color="Green" title="E31">A3</font></i>: 
not <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:51"/>
( <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> )
 
<b>by </b><i><a class="txt" href="fintopo6.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="ref" href="fintopo6.html#D4" target="_self" title="FINTOPO6:def.4">Def4</a></i>;<br/>
<a NAME="E5:51"/><b>then </b><i><font color="Green" title="E32">A4</font></i>: 
<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> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="fintopo6.html#E3:51"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="fintopo6.html#T34" target="_self" title="FINTOPO6:th.34">Th34</a></i>;<br/>
<a NAME="E6:51"/>
( <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E7:51"/><b>then </b>
( <font color="Maroon" title="c2">A</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> &amp; <font color="Maroon" title="c3">B</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>by </b><i><a class="txt" href="fintopo6.html#E1:51"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="fintopo6.html#E5:51"><i><font color="Green" title="E32">A4</font></i></a>, <a class="ref" href="fintopo6.html#D4" target="_self" title="FINTOPO6:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:51"/>
contradiction
 <b>by </b><i><a class="txt" href="fintopo6.html#E2:51"><i><font color="Green" title="E30">A2</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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