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<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><a NAME="E1:3"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  holds <br/>(  not <font color="Olive">A</font> is <a href="connsp_1.html#V2">connected</a> or  not <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> )
 ;<br/>

<b>consider </b><font color="Maroon">F</font> being   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:3"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">A</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">A</font> ) )
 <b>and </b><br/><a NAME="E4:3"/><i><font color="Green">E25</font></i>: 
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><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">F</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E3:3_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> ;<br/><a NAME="E5:3_1"/>
( <font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:3_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:3"/>
 <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T2">ZFMISC_1:2</a></i>;<br/>


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