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<b>let </b><font color="Maroon" title="c1">GX</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</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>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font> &amp; <font color="Maroon" title="c3">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c3">B</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>  &amp; <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>assume </b><a NAME="E2:18"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon" title="c4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E4:18"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font></span>)</span>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E15">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="connsp_3.html#E4:18"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E6:18"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="connsp_3.html#T7" target="_self" title="CONNSP_3:th.7">Th7</a></i>;<br/>
<a NAME="E7:18"/>
<font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="connsp_1.html#D5" title="CONNSP_1:def.5">CONNSP_1:def 5</a></i>;<br/>
<a NAME="E8:18"/><b>then </b><i><font color="Green" title="E17">A5</font></i>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E5:18"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="connsp_3.html#T15" target="_self" title="CONNSP_3:th.15">Th15</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green" title="E18">A6</font></i>: 
 <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <span class="p1">(<span class="default"><a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="connsp_3.html#T11" target="_self" title="CONNSP_3:th.11">Th11</a></i>;<br/>
<a NAME="E10:18"/>
 <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="connsp_3.html#T5" target="_self" title="CONNSP_3:th.5">Th5</a></i>;<br/>
<a NAME="E11:18"/><b>then </b><i><font color="Green" title="E19">A7</font></i>: 
<span class="p1">(<span class="default"><a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="connsp_3.html#E5:18"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="connsp_3.html#E6:18"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="connsp_3.html#E8:18"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="connsp_3.html#E9:18"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="connsp_3.html#T15" target="_self" title="CONNSP_3:th.15">Th15</a></i>;<br/>
<a NAME="E12:18"/>
<font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="connsp_3.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="connsp_3.html#T1" target="_self" title="CONNSP_3:th.1">Th1</a></i>;<br/>
<b>hence </b><a NAME="E13:18"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:18"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="connsp_3.html#E11:18"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>


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