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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>; <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>, <font color="Olive" title="b2">C</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">V</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Olive" title="b2">C</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp;  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Olive" title="b2">C</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="Olive" title="b1">V</font></span><br/><b>let </b><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">C</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">V</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c3">C</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp;  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> implies <font color="Maroon" title="c3">C</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">V</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:7"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">V</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E2:7"/><i><font color="Green" title="E7">A2</font></i>: 
<font color="Maroon" title="c3">C</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="connsp_3.html#K1" title="CONNSP_3:func.1">Component_of</a> <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> or <font color="Maroon" title="c3">C</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">V</font> )</span><br/>

<b>assume </b><a NAME="E3:7"/><i><font color="Green" title="E8">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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">C</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">V</font></span><br/>

<b>consider </b><font color="Maroon" title="c4">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="E5:7"/><i><font color="Green" title="E9">A4</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="c4">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="E6:7"/><i><font color="Green" title="E10">A5</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="c4">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/>
<a NAME="E7:7"/>
<font color="Maroon" title="c2">V</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="c2">V</font>
 
<b>by </b><i><a class="txt" href="#E1:7"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="connsp_3.html#T1" target="_self" title="CONNSP_3:th.1">Th1</a></i>;<br/>
<a NAME="E8:7"/><b>then </b>
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E9:7"/><b>then </b>
<font color="Maroon" title="c3">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="txt" href="#E2:7"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="#E5:7"><i><font color="Green" title="E9">A4</font></i></a></i>;<br/>
<a NAME="E10:7"/><b>then </b>
<font color="Maroon" title="c3">C</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="c2">V</font>
 
<b>by </b><i><a class="txt" href="#E6:7"><i><font color="Green" title="E10">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></i>;<br/>
<b>hence </b><a NAME="E11:7"/>
<font color="Maroon" title="c3">C</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">V</font>
 <b>by </b><i><a class="txt" href="#E3:7"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</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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