<?xml version="1.0"?>
<div class="add">

<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> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>;<br/>



<a NAME="E1:25"/><i><font color="Green" title="E16">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="fintopo6.html#D3" target="_self" title="FINTOPO6:def.3">Def3</a></i>;<br/>
<b>thus </b><a NAME="E2:25"/>
( <font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a> implies <font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>
 ;<br/>

<a NAME="E2:25_1"/>
for <font color="Olive" title="b1">B2</font>, <font color="Olive" title="b2">C2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>  st  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">B2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">C2</font> &amp; <font color="Olive" title="b1">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">C2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b1">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Olive" title="b2">C2</font> holds <br/><font color="Olive" title="b1">B2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">C2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">B2</font>, <font color="Maroon" title="c4">C2</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green" title="E18">A3</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B2</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C2</font> &amp; <font color="Maroon" title="c3">B2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">C2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C2</font> )
 ;<br/>

the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">A</font>
<b>by </b><i><a class="ref" href="fintopo6.html#D3" target="_self" title="FINTOPO6:def.3">Def3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">B3</font> = <font color="Maroon" title="c3">B2</font>, <font color="Maroon" title="c6">C3</font> = <font color="Maroon" title="c4">C2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font> <b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:25_1_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c5">B3</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">C3</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E1:25"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="fintopo6.html#E1:25_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="fintopo6.html#E1:25_1_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a></i>;<br/>
<a NAME="E5:25_1_1"/><i><font color="Green" title="E20">A5</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">B3</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C2</font>
<b>by </b><i><a class="ref" href="fintopo6.html#T13" target="_self" title="FINTOPO6:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">B3</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">C3</font>
<b>by </b><i><a class="txt" href="fintopo6.html#E5:25_1_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
;<br/>
<a NAME="E7:25_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="fintopo6.html#E4:25_1_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1"/>
<font color="Maroon" title="c3">B2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c4">C2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:25_1"/><b>then </b>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> is <a href="fin_topo.html#V4" title="FIN_TOPO:attr.4">connected</a>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1"/>
<font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="fintopo6.html#D1" target="_self" title="FINTOPO6:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E3:25"/>
<font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font> is <a href="fintopo6.html#V1" title="FINTOPO6:attr.1">connected</a>
 ;<br/>

<a NAME="E4:25"/><b>then </b><i><font color="Green" title="E17">A6</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> 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>let </b><font color="Maroon" title="c3">B</font>, <font color="Maroon" title="c4">C</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">FT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a><br/>

<b>assume </b><a NAME="E5:25"/><i><font color="Green" title="E18">A7</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c3">B</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">B</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c4">C</font> )
 ;<br/>

<a NAME="E6:25"/><i><font color="Green" title="E19">A8</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="ref" href="fintopo6.html#D3" target="_self" title="FINTOPO6:def.3">Def3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">B2</font> = <font color="Maroon" title="c3">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="txt" href="fintopo6.html#E5:25"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">C2</font> = <font color="Maroon" title="c4">C</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a>, <a class="txt" href="fintopo6.html#E5:25"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="fintopo6.html#E6:25"><i><font color="Green" title="E19">A8</font></i></a></i>;<br/>
<a NAME="E9:25"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c5">B2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c6">C2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E5:25"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="fintopo6.html#E4:25"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="fintopo6.html#E6:25"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="fin_topo.html#D18" title="FIN_TOPO:def.18">FIN_TOPO:def 18</a></i>;<br/>
<a NAME="E10:25"/><i><font color="Green" title="E21">A10</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">C2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E11:25"/>
<font color="Maroon" title="c5">B2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">FT</font> <a href="fintopo6.html#K1" title="FINTOPO6:func.1">|</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="fintopo6.html#T13" target="_self" title="FINTOPO6:th.13">Th13</a></i>;<br/>
<a NAME="E12:25"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">B2</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">C2</font>
 
<b>by </b><i><a class="txt" href="fintopo6.html#E10:25"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E13:25"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">B</font> <a href="fin_topo.html#K9" title="FIN_TOPO:func.9">^b</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c4">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="fintopo6.html#E9:25"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>


</div>
