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

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



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a>
 ;<br/>

<b>thus </b><a NAME="E2:12"/>
( <font color="Maroon">B</font> is <a href="fin_topo.html#V5">connected</a> implies for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Olive">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">B</font> is <a href="fin_topo.html#V5">connected</a>
 ;<br/>

<a NAME="E2:12_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Olive">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> &amp; not <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:12_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Maroon">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> )
 ;<br/>

<i><font color="Green">E30</font></i>: <font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font></span>)</span> = 
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T39">XBOOLE_1:39</a></i>
<br/>.= 
<font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="xboole_1.html#T12">XBOOLE_1:12</a></i>
;<br/>
<a NAME="E3:12_1_1"/>
( <font color="Maroon">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font></span>)</span> <a href="fin_topo.html#K9">^b</a>  )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:12_1_1"/><b>then </b>
<font color="Maroon">C</font>,<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> <a href="fintopo4.html#R1" target="_self">are_separated</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E5:12_1_1"/><b>then </b>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E6:12_1_1"/>
( <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a>, <a class="ref" href="xboole_1.html#T38">XBOOLE_1:38</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:12_1"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Olive">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> )
 ;<br/>


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

<b>thus </b><a NAME="E3:12"/>
( ( for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Olive">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> ) ) implies <font color="Maroon">B</font> is <a href="fin_topo.html#V5">connected</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2"/>
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  holds <br/>(  not <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  or  not <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> or  not <font color="Olive">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> )
 ;<br/>

<a NAME="E2:12_2"/><b>then </b><i><font color="Green">E31</font></i>: 
for <font color="Olive">C</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> &amp; <font color="Olive">C</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> &amp; not <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Olive">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E3:12_2"/>
for <font color="Olive">A</font>, <font color="Olive">B2</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B2</font> &amp; <font color="Olive">A</font>,<font color="Olive">B2</font> <a href="fintopo4.html#R1" target="_self">are_separated</a>  &amp; not <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> holds <br/><font color="Olive">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>, <font color="Maroon">B2</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:12_2_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">A</font>,<font color="Maroon">B2</font> <a href="fintopo4.html#R1" target="_self">are_separated</a>  )
 ;<br/>

<a NAME="E2:12_2_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E3:12_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B2</font> <a href="fin_topo.html#K9">^b</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:12_2_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B2</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>;<br/>
<a NAME="E5:12_2_1"/>
<font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B2</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E6:12_2_1"/><b>then </b>
<font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E7:12_2_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:12_2_1"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B2</font> or <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E9:12_2_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">B2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:12_2"/>
<font color="Maroon">B</font> is <a href="fin_topo.html#V5">connected</a>
 <b>by </b><i>, </i>;<br/>


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


</div>
