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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">C</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="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; <font color="Olive" title="b1">C</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>  holds <br/>for <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>  holds <br/>(  not <font color="Olive" title="b2">S</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> or <font color="Olive" title="b1">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">S</font> or <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">S</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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; <font color="Maroon" title="c2">C</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>  implies for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>  holds <br/>(  not <font color="Olive" title="b1">S</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> or <font color="Maroon" title="c2">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">S</font> or <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:52"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
( <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; <font color="Maroon" title="c2">C</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>  holds <br/>(  not <font color="Olive" title="b1">S</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> or <font color="Maroon" title="c2">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b1">S</font> or <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">S</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">S</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">S</font> <a href="fintopo6.html#R1" title="FINTOPO6:pred.1">is_a_component_of</a> <font color="Maroon" title="c1">FT</font> or <font color="Maroon" title="c2">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:52"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c3">S</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">C</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">S</font> or <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font> )</span><br/>

<a NAME="E3:52"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:52"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font></span><br/>

<a NAME="E5:52"/>
<font color="Maroon" title="c3">S</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="fintopo6.html#D4" target="_self" title="FINTOPO6:def.4">Def4</a></span>;<br/>
<a NAME="E6:52"/><span class="kw">then </span>
<font color="Maroon" title="c2">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font> is  <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:52"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="ref" href="fintopo6.html#T34" target="_self" title="FINTOPO6:th.34">Th34</a>, <a class="ref" href="fintopo4.html#T6" title="FINTOPO4:th.6">FINTOPO4:6</a></span>;<br/>
<a NAME="E7:52"/><span class="kw">then </span>
<font color="Maroon" title="c3">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E3:52"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="fintopo6.html#D4" target="_self" title="FINTOPO6:def.4">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:52"/>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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