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<b>let </b><font color="Maroon" title="c1">GX</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">B</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st <font color="Olive" title="b1">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="Olive" title="b2">B</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; not <font color="Olive" title="b1">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">B</font> holds <br/><font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font> &amp; not <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>





<b>assume </b><a NAME="E1:40"/><i><font color="Green" title="E33">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> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">GX</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )</span><br/>

<a NAME="E2:40"/>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font> implies <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:40_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E2:40_1"/><i><font color="Green" title="E35">A3</font></i>: 
not <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:40_1"/>
( <font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <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_1.html#E1:40"><i><font color="Green" title="E33">A1</font></i></a>, <a class="ref" href="connsp_1.html#D5" target="_self" title="CONNSP_1:def.5">Def5</a></i>;<br/>
<a NAME="E4:40_1"/><b>then </b><i><font color="Green" title="E36">A4</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</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_1.html#E2:40_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="connsp_1.html#T18" target="_self" title="CONNSP_1:th.18">Th18</a></i>;<br/>
<a NAME="E5:40_1"/>
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E6:40_1"/><b>then </b>
( <font color="Maroon" title="c2">A</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> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</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> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="txt" href="connsp_1.html#E1:40"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="connsp_1.html#E4:40_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="ref" href="connsp_1.html#D5" target="_self" title="CONNSP_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:40_1"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:40_1"><i><font color="Green" title="E34">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font> or <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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