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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">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  st ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  holds <br/>(  not <font color="Olive" title="b2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> or  not <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">A</font> ) ) holds <br/> <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Olive" title="b1">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/><b>let </b><font color="Maroon" title="c2">V</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"> ( ( for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  holds <br/>(  not <font color="Olive" title="b1">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> or  not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) ) implies  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>



<b>assume </b><a NAME="E1:3"/><i><font color="Green" title="E3">A1</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>  holds <br/>(  not <font color="Olive" title="b1">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> or  not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<b>consider </b><font color="Maroon" title="c3">F</font> being   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E3:3"/><i><font color="Green" title="E4">A2</font></i>: 
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> holds <br/> ( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> iff ( <font color="Olive" title="b1">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">A</font> ) )
 <b>and </b><br/><a NAME="E4:3"/><i><font color="Green" title="E5">A3</font></i>: 
 <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="connsp_3.html#D1" target="_self" title="CONNSP_3:def.1">Def1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:3_1"/>
<font color="Maroon" title="c3">F</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><b>then consider </b><font color="Maroon" title="c4">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font><b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c4">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10" title="SUBSET_1:th.10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">A</font> = <font color="Maroon" title="c4">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font> ;<br/><a NAME="E5:3_1"/>
( <font color="Maroon" title="c5">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">A</font> )
 <b>by </b><i><a class="txt" href="connsp_3.html#E3:3"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="connsp_3.html#E3:3_1"><i><font color="Green" title="E6">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E6:3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="connsp_3.html#E1:3"><i><font color="Green" title="E3">A1</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="E6:3"/>
 <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">V</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#E4:3"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</a></i>; <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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