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<b>let </b><font color="Maroon">T</font> be  <a href="connsp_1.html#V1">connected</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be  <a href="pre_topc.html#V3">open</a> <a href="pre_topc.html#V4">closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:62"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font>
 ;<br/>

<a NAME="E3:62"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="borsuk_5.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E4:62"/><i><font color="Green">E49</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T18">PRE_TOPC:18</a></i>;<br/>
<a NAME="E5:62"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<a NAME="E6:62"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">A</font>,<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E6:62"><i><font color="Green">E50</font></i></a>, <a class="ref" href="connsp_1.html#T3">CONNSP_1:3</a></i>;<br/>
<a NAME="E7:62"/>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E3:62"><i><font color="Green">E48</font></i></a>, <a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/>
<a NAME="E8:62"/><b>then </b>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E9:62"/><b>then </b>
not  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="borsuk_5.html#E4:62"><i><font color="Green">E49</font></i></a>, <a class="txt" href="borsuk_5.html#E6:62"><i><font color="Green">E50</font></i></a>, <a class="ref" href="borsuk_5.html#T11" target="_self">Th11</a>, <a class="ref" href="connsp_1.html#T16">CONNSP_1:16</a></i>;<br/>
<b>hence </b><a NAME="E10:62"/>
contradiction
 <b>by </b><i><a class="ref" href="connsp_1.html#T28">CONNSP_1:28</a></i>;<br/>


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