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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:19"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E2:19"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E3:19"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/>

<b>assume </b><a NAME="E4:19"/>
( not <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E5:19"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="connsp_1.html#E2:19"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/>
<a NAME="E6:19"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></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/>
<a NAME="E7:19"/><b>then </b><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E8:19"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E9:19"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 
<b>by </b><i><a class="txt" href="connsp_1.html#E3:19"><i><font color="Green">E19</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="connsp_1.html#E2:19"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E11:19"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:19"><i><font color="Green">E17</font></i></a>, <a class="txt" href="connsp_1.html#E6:19"><i><font color="Green">E20</font></i></a>, <a class="txt" href="connsp_1.html#E7:19"><i><font color="Green">E21</font></i></a>, <a class="txt" href="connsp_1.html#E9:19"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T16" target="_self">Th16</a></i>;<br/>


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