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<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:58"/><i><font color="Green" title="E36">A1</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>and </b><br/><a NAME="E2:58"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>
 ;<br/>

<a NAME="E3:58"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E1:58"><i><font color="Green" title="E36">A1</font></i></a></i>;<br/>
<a NAME="E4:58"/><i><font color="Green" title="E39">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18" title="PRE_TOPC:th.18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E5:58"/>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/>
<a NAME="E6:58"/><b>then </b><i><font color="Green" title="E40">A5</font></i>: 
<font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E4:58"><i><font color="Green" title="E39">A4</font></i></a>, <a class="ref" href="connsp_1.html#T3" title="CONNSP_1:th.3">CONNSP_1:3</a></i>;<br/>
<a NAME="E7:58"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E2:58"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="pre_topc.html#T23" title="PRE_TOPC:th.23">PRE_TOPC:23</a></i>;<br/>
<a NAME="E8:58"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:58"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="borsuk_5.html#E4:58"><i><font color="Green" title="E39">A4</font></i></a>, <a class="txt" href="borsuk_5.html#E6:58"><i><font color="Green" title="E40">A5</font></i></a>, <a class="ref" href="connsp_1.html#T16" title="CONNSP_1:th.16">CONNSP_1:16</a></i>;<br/>
<b>hence </b><a NAME="E9:58"/>
contradiction
 <b>by </b><i><a class="ref" href="connsp_1.html#T28" title="CONNSP_1:th.28">CONNSP_1:28</a></i>;<br/>


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