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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">C</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>let </b><font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">GX</font>;<br/>





<b>assume </b><a NAME="E1:48"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c2">C</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ;<br/>

<b>assume </b><a NAME="E2:48"/><i><font color="Green" title="E40">A2</font></i>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c3">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">C</font>
 ;<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="E4:48"/><i><font color="Green" title="E41">A3</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="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> ) )
 <b>and </b><br/><a NAME="E5:48"/><i><font color="Green" title="E42">A4</font></i>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c3">x</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_1.html#D7" target="_self" title="CONNSP_1:def.7">Def7</a></i>;<br/>
<a NAME="E6:48"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40" target="_self" title="CONNSP_1:th.40">Th40</a></i>;<br/>
<a NAME="E7:48"/><b>then </b>
<font color="Maroon" title="c2">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="connsp_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="connsp_1.html#E2:48"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="connsp_1.html#E4:48"><i><font color="Green" title="E41">A3</font></i></a></i>;<br/>
<a NAME="E8:48"/><b>then </b>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E5:48"><i><font color="Green" title="E42">A4</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="E9:48"/>
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:48"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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