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

<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font><b> such that </b><a NAME="E1:34_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> holds  <font color="Maroon">a</font>,<font color="Olive">x</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/><a NAME="E2:34_1"/>
<font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E9:34_1"><i><font color="Green">E30</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">C1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E4:34_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">C1</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E5:34_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 <b>and </b><br/><a NAME="E6:34_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E7:34_1"/>
<font color="Maroon">a</font>,<font color="Maroon">y</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E9:34_1"><i><font color="Green">E30</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">C2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E9:34_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">C2</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E10:34_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 <b>and </b><br/><a NAME="E11:34_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E12:34_1"/>
<font color="Maroon">C1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">C2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">GX</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E11:34_1"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E13:34_1"/><b>then </b>
<font color="Maroon">C1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C2</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E14:34_1"/><b>then </b>
not <font color="Maroon">C1</font>,<font color="Maroon">C2</font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i/>;<br/><a NAME="E15:34_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E10:34_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="connsp_1.html#T6" target="_self">Th6</a>, </i>;<br/><a NAME="E16:34_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">C2</font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E15:34_1"><i><font color="Green">E33</font></i></a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E17:34_1"/>
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 <b>by </b><i>, <a class="ref" href="connsp_1.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 ;<br/><b>consider </b><font color="Maroon">a</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/><a NAME="E3:34_2"/>
for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> holds  <font color="Maroon">a</font>,<font color="Olive">y</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:34_2"/>
 ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> st <br/>for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="connsp_1.html#R2" target="_self">are_joined</a> 
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
(  ex <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> st <br/>for <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="connsp_1.html#R2" target="_self">are_joined</a>  iff for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font> holds  <font color="Olive">x</font>,<font color="Olive">y</font> <a href="connsp_1.html#R2" target="_self">are_joined</a>  )
 <b>by </b><i/>;<br/>


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