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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">filled</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/>

<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
( <font color="Maroon" title="c1">FT</font> is  <a href="orders_2.html#V3" title="ORDERS_2:attr.3">filled</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E2:40"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E3:40"/>
 not <font color="Maroon" title="c1">FT</font> is  <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:40"/><span class="kw">then </span>
 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> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:40"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E7:40"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E8:40"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E9:40"/><span class="lab"><font color="Green" title="E22">A6</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="fintopo6.html#T7" target="_self" onmouseover="rs('fintopo6/T7')" onmouseout="rh()">Th6</a></span>;<br/>
<a NAME="E10:40"/>
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E8:40"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="fintopo6.html#T28" target="_self" onmouseover="rs('fintopo6/T28')" onmouseout="rh()">Th27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:40"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E6:40"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E9:40"><span class="lab"><font color="Green" title="E22">A6</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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