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



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">FT</font> is <a href="fin_topo.html#V2">symmetric</a>
 ;<br/>

<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1"/>
<font color="Maroon">IT</font> is <a href="fin_topo.html#V5">connected</a>
 ;<br/><a NAME="E2:11_1"/><b>then </b><i><font color="Green">E28</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/><font color="Olive">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D18">FIN_TOPO:def 18</a></i>;<br/><b>thus </b><a NAME="E3:11_1"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font>,<font color="Olive">B</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">IT</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_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">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:11_1_1"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="fintopo4.html#R1" target="_self">are_separated</a>  )
 ;<br/>

<a NAME="E2:11_1_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_1_1_1_1"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:11_1_1_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:11_1_1_1_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:11_1_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11_1_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font>,<font color="Olive">B</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">IT</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font>
 ;<br/><a NAME="E2:11_2"/>
for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Olive">B</font> holds <br/><font color="Olive">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_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">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>;<br/>

<b>assume </b><a NAME="E1:11_2_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</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="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_1_1"/><i><font color="Green">E44</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">B</font>
 ;<br/><a NAME="E2:11_2_1_1"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</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/><a NAME="E3:11_2_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a> 
 <b>by </b><i><a class="ref" href="fin_topo.html#T18">FIN_TOPO:18</a></i>;<br/><a NAME="E4:11_2_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a> </span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T26">XBOOLE_1:26</a></i>;<br/><a NAME="E5:11_2_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</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#T3">XBOOLE_1:3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:11_2_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a> 
 ;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:11_2_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> <a href="fin_topo.html#K9">^b</a>  )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font><b> such that </b><br/><a NAME="E5:11_2_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font> )
 <b>by </b><i/>;<br/><a NAME="E6:11_2_1_1_1"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</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>consider </b><font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>;<br/><a NAME="E8:11_2_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon">y2</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT</font> ;<br/><a NAME="E10:11_2_1_1_1"/>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font>
 <b>by </b><i>, , <a class="ref" href="fin_topo.html#D15">FIN_TOPO:def 15</a></i>;<br/><a NAME="E11:11_2_1_1_1"/><b>then </b>
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E12:11_2_1_1_1"/><b>then </b>
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E13:11_2_1_1_1"/><b>then </b>
<font color="Maroon">y2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a> 
 ;<br/><b>hence </b><a NAME="E14:11_2_1_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><a NAME="E7:11_2_1_1"/><b>then </b>
<font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="fintopo4.html#R1" target="_self">are_separated</a> 
 <b>by </b><i>, </i>;<br/><a NAME="E8:11_2_1_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E9:11_2_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:11_2_1"/>
<font color="Maroon">A</font> <a href="fin_topo.html#K9">^b</a>  <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 ;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:11_2"/>
<font color="Maroon">IT</font> 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/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:11"/>
( <font color="Maroon">IT</font> is <a href="fin_topo.html#V5">connected</a> iff for <font color="Olive">A</font>, <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">FT</font>  st <font color="Maroon">IT</font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">B</font> &amp; <font color="Olive">A</font>,<font color="Olive">B</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">IT</font> holds <br/><font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">IT</font> )
 <b>by </b><i/>;<br/>


</div>
