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<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>, <font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><a NAME="E1:16"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<a NAME="E2:16"/>
for <font color="Olive">C</font>, <font color="Olive">D</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>  st <font color="Olive">C</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Olive">D</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Olive">C</font> <a href="subset_1.html#K9">/\</a> <font color="Olive">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">C</font> <a href="subset_1.html#K4">\/</a> <font color="Olive">D</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Olive">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>, <font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">C</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">D</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">C</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">D</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
;<br/>
<a NAME="E3:16_1"/>
<font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b>
not <font color="Maroon">C</font>,<font color="Maroon">D</font> <a href="connsp_1.html#R1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="connsp_1.html#T2">CONNSP_1:2</a></i>;<br/>
<a NAME="E5:16_1"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">D</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T18">CONNSP_1:18</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">C</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/><b>then </b><i><font color="Green">E26</font></i>: 
( <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:16"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:16"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_3.html#K1" target="_self">Component_of</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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