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<b>let </b><font color="Maroon">GX</font> be  non <a href="struct_0.html#V3">empty</a> <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>, <font color="Maroon">V</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="E1:10"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">V</font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E3:10"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font>
 <b>and </b><br/><a NAME="E4:10"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>assume </b><a NAME="E5:10"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> ;<br/>
<a NAME="E7:10"/>
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:10"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T67">XBOOLE_1:67</a></i>;<br/>
<a NAME="E9:10"/>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E10:10"/><b>then </b>
<font color="Maroon">A'</font>, <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i>, <a class="txt" href="connsp_3.html#E2:10"><i><font color="Green">E13</font></i></a>, <a class="ref" href="connsp_1.html#T36">CONNSP_1:36</a></i>;<br/>
<b>hence </b><a NAME="E11:10"/>
contradiction
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T2">CONNSP_1:2</a></i>;<br/>


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