<?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">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 ;<br/><a NAME="E2:9_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">connected</a> )
 <b>by </b><i><a class="ref" href="connsp_1.html#T34">CONNSP_1:34</a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/><b>take </b><font color="Maroon">V</font> = <font color="Maroon">A</font>;<br/><b>thus </b><a NAME="E3:9_1"/>
( <font color="Maroon">V</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><a NAME="E1:9_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">V</font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E2:9_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E3:9_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">V</font>
 ;<br/><a NAME="E4:9_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, , , </i>;<br/><a NAME="E5:9_2"/>
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">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="E6:9_2"/>
<font color="Maroon">A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font>
 <b>by </b><i>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:9"/>
( <font color="Maroon">A</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">GX</font> iff  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> st <br/>( <font color="Olive">V</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Olive">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Olive">V</font> ) )
 <b>by </b><i/>;<br/>


</div>
