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





<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:27"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 ;<br/>

<a NAME="E5:27"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span> is <a href="connsp_1.html#V1" target="_self">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E2:27"><i><font color="Green">E24</font></i></a>, </i>;<br/>
<a NAME="E6:27"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>set </b><font color="Maroon">AB</font> = <font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font>;<br/>
<a NAME="E7:27"/><i><font color="Green">E31</font></i>: 
( <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>then reconsider </b><font color="Maroon">A1</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E4:27"><i><font color="Green">E27</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">B1</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E4:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green">E32</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E4:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xboole_1.html#T55">XBOOLE_1:55</a></i>;<br/>
<a NAME="E11:27"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">B1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E4:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E14:27"/>
<font color="Maroon">B</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E4:27"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E15:27"/><b>then </b>
( <font color="Maroon">A1</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">B1</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i><a class="txt" href="connsp_1.html#E1:27"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="txt" href="connsp_1.html#E10:27"><i><font color="Green">E32</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">A1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">B1</font>,<font color="Maroon">B1</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">A1</font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E17:27"/>
<font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B1</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E3:27"><i><font color="Green">E25</font></i></a>, <a class="ref" href="connsp_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E18:27"/><b>then </b>
( <font color="Maroon">A1</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">B1</font> is <a href="connsp_1.html#V2" target="_self">connected</a> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#T4" target="_self">Th4</a>, <a class="txt" href="connsp_1.html#E5:27"><i><font color="Green">E29</font></i></a>, <a class="txt" href="connsp_1.html#E6:27"><i><font color="Green">E30</font></i></a>, <a class="txt" href="connsp_1.html#E7:27"><i><font color="Green">E31</font></i></a>, <a class="txt" href="connsp_1.html#E11:27"><i><font color="Green">E33</font></i></a>, <a class="ref" href="connsp_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>hence </b><a NAME="E19:27"/>
( <font color="Maroon">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a> )
 <b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a></i>;<br/>


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