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<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E3:20"/><i><font color="Green">E25</font></i>: 
not <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/>

<b>assume </b><a NAME="E4:20"/>
not <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E6:20"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E7:20"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">P</font>,<font color="Maroon">Q</font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>and </b><br/><a NAME="E8:20"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E9:20"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/>
<a NAME="E10:20"/>
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E1:20"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E11:20"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</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="E12:20"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E2:20_1"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E3:20_1"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E4:20_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:20_1"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:20"><i><font color="Green">E24</font></i></a>, <a class="txt" href="connsp_1.html#E3:20"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:20"/><i><font color="Green">E47</font></i>: 
(  not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> or  not <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#D1" target="_self">Def1</a>, <a class="txt" href="connsp_1.html#E1:20"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<a NAME="E15:20"/><i><font color="Green">E48</font></i>: 
(  not <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> or  not <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#D1" target="_self">Def1</a>, <a class="txt" href="connsp_1.html#E1:20"><i><font color="Green">E22</font></i></a>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_2"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 ;<br/><a NAME="E2:20_2"/><b>then </b>
( <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/><a NAME="E3:20_2"/><b>then </b>
<font color="Maroon">Q</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E4:20_2"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><b>hence </b><a NAME="E5:20_2"/>
contradiction
 <b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="txt" href="connsp_1.html#E3:20"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E17:20"/>
contradiction
 <b>by </b><i>, , <a class="txt" href="connsp_1.html#E1:20"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T4" target="_self">Th4</a>, <a class="txt" href="connsp_1.html#E6:20"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a>, <a class="txt" href="connsp_1.html#E7:20"><i><font color="Green">E29</font></i></a>, </i>;<br/>


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