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<b>let </b><font color="Maroon">FT</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>



<a NAME="E1:25"/><i><font color="Green">E27</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:25"/>
( <font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a> implies <font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font> is <a href="fintopo6.html#V1" target="_self">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">E31</font></i>: 
<font color="Maroon">A</font> is <a href="fin_topo.html#V5">connected</a>
 ;<br/>

<a NAME="E2:25_1"/>
for <font color="Olive">B2</font>, <font color="Olive">C2</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span>  st  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">B2</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">C2</font> &amp; <font color="Olive">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B2</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">C2</font> holds <br/><font color="Olive">B2</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">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">B2</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">C2</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:25_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">B2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C2</font> )
 ;<br/>

<i><font color="Green">E41</font></i>: the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> = 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span>

<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i/>
;<br/>
<b>reconsider </b><font color="Maroon">B3</font> = <font color="Maroon">B2</font>, <font color="Maroon">C3</font> = <font color="Maroon">C2</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:25_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">B3</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C3</font>
 
<b>by </b><i>, , , <a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/>
<a NAME="E5:25_1_1"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">B3</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p3">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">B3</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C3</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/>
<a NAME="E7:25_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1"/>
<font color="Maroon">B2</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:25_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> is <a href="fin_topo.html#V5">connected</a>
 
<b>by </b><i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/>
<b>hence </b><a NAME="E4:25_1"/>
<font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font> is <a href="fintopo6.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/>


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

<b>assume </b><a NAME="E3:25"/>
<font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font> is <a href="fintopo6.html#V1" target="_self">connected</a>
 ;<br/>

<a NAME="E4:25"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> is <a href="fin_topo.html#V5">connected</a>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a><br/>

<b>assume </b><a NAME="E5:25"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E6:25"/><i><font color="Green">E81</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">B2</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i>, , <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">C2</font> = <font color="Maroon">C</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, , </i>;<br/>
<a NAME="E9:25"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">B2</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C2</font>
 
<b>by </b><i>, , , <a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/>
<a NAME="E10:25"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C2</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E11:25"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">B2</font> <a href="fin_topo.html#K9">^b</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">FT</font> <a href="fintopo6.html#K1" target="_self">|</a> <font color="Maroon">A</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:25"/>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">B2</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<b>hence </b><a NAME="E13:25"/>
<span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>


</div>
