<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">GX</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>



<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="connsp_1.html#R3" target="_self">is_a_component_of</a> <font color="Maroon">GX</font>
 ;<br/><a NAME="E2:49_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font> &amp; <font color="Maroon">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E7:49_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="connsp_1.html#E9:49_1"><i><font color="Green">E31</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E4:49_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T41">PRE_TOPC:41</a></i>;<br/><b>take </b><font color="Maroon">x</font> = <font color="Maroon">y</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="E6:49_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> holds <br/> ( <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> iff ( <font color="Olive">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">B</font> ) )
 <b>and </b><br/><a NAME="E7:49_1"/><i><font color="Green">E30</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1" target="_self">Component_of</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E8:49_1"/>
<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="connsp_1.html#T7" target="_self">Th7</a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a>, <a class="ref" href="connsp_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E9:49_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E10:49_1"/>
 <a href="connsp_1.html#K1" target="_self">Component_of</a> <font color="Maroon">x</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E11:49_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1" target="_self">Component_of</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T10" target="_self">Th10</a>, , <a class="txt" href="connsp_1.html#E7:49_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font><b> such that </b><a NAME="E1:49_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1" target="_self">Component_of</a> <font color="Maroon">x</font>
 ;<br/><a NAME="E2:49_2"/>
( <font color="Maroon">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; ( for <font color="Olive">B</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> holds <br/><font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> ) )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E3:49_2"/>
<font color="Maroon">A</font> <a href="connsp_1.html#R3" target="_self">is_a_component_of</a> <font color="Maroon">GX</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
( <font color="Maroon">A</font> <a href="connsp_1.html#R3" target="_self">is_a_component_of</a> <font color="Maroon">GX</font> iff  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> st <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="connsp_1.html#K1" target="_self">Component_of</a> <font color="Olive">x</font> )
 <b>by </b><i/>;<br/>


</div>
