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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">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<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>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">C</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E3:21"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font>
 ;<br/>

<b>assume </b><a NAME="E4:21"/>
not <font color="Maroon">A</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="E6:21"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</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="E7:21"/><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="E8:21"/><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> &amp; <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="E9:21"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, </i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green">E31</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <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>  &amp; <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span> <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/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:21_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E3:21_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E4:21_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <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="txt" href="connsp_1.html#E3:21"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E5:21_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K9">/\</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#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E6:21_1"/><b>then </b>
<font color="Maroon">A</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="txt" href="connsp_1.html#E6:21"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E7:21_1"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:21"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:21"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_2"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:21_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E3:21_2"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">Q</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E4:21_2"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:21"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E5:21_2"/>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E6:21_2"/><b>then </b>
<font color="Maroon">P</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E6:21"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><b>hence </b><a NAME="E7:21_2"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:21"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E2:21"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T21">XBOOLE_1:21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E13:21"/>
contradiction
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D1" target="_self">Def1</a>, <a class="txt" href="connsp_1.html#E1:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="connsp_1.html#T4" target="_self">Th4</a>, </i>;<br/>


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