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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">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><a NAME="E1:31"/>
not <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> 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="E3:31"/><i><font color="Green">E22</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> <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="E4:31"/><i><font color="Green">E24</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="E5:31"/><i><font color="Green">E25</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="E6:31"/><i><font color="Green">E27</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="E7:31"/>
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
not <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green">E29</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> <font color="Maroon">P</font>
 
;<br/>
<a NAME="E3:31_1"/>
<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#E6:31"><i><font color="Green">E27</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E4:31_1"/>
contradiction
 <b>by </b><i><a class="ref" href="connsp_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:31"/>
contradiction
 <b>by </b><i><a class="ref" href="connsp_1.html#T4" target="_self">Th4</a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a>, , <a class="ref" href="zfmisc_1.html#T44">ZFMISC_1:44</a></i>;<br/>


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