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<b>let </b><font color="Maroon">G</font> be   <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">G</font>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>, <font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">G</font>;<br/>



<b>assume </b><a NAME="E1:106"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">B</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">G</font> &amp; <font color="Maroon">C</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">B</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E2:106"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">G</font> or <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/>
<a NAME="E3:106"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">C</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> or <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/>
<a NAME="E4:106"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font>,<font color="Maroon">B</font> <a href="connsp_1.html#R1">are_separated</a>  )
 
<b>by </b><i><a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, <a class="ref" href="connsp_1.html#T36">CONNSP_1:36</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:106_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> or <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font> )
 </span><b>by </b><i>, <a class="ref" href="connsp_1.html#T2">CONNSP_1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_1"><b>suppose </b></a><a NAME="E1:106_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:106_1_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="106_1_2"><b>suppose </b></a><a NAME="E1:106_1_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/><div class="add"><a NAME="E2:106_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</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="E3:106_1_2"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, <a class="ref" href="jordan2c.html#T18" target="_self">Th18</a>, , <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E4:106_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T21" target="_self">Th21</a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/><a NAME="E5:106_1_2"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>hence </b><a NAME="E6:106_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#D4" target="_self">Def4</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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