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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>







<div><i><font color="Green">E21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:23_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>and </b><br/><a NAME="E2:23_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E3:23_1"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E4:23_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:23_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E2:23_1_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/><i><font color="Green">E28</font></i>:  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
;<br/><a NAME="E4:23_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:23_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:23_1_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:23_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:23_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><a NAME="E6:23_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E4:23_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E7:23_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E8:23_1_1_2"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E9:23_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E7:23_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E10:23_1_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:23_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E3:23_1_1_3"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E4:23_1_1_3"/>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E5:23_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><a NAME="E6:23_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E4:23_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a></i>;<br/><a NAME="E7:23_1_1_3"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T9" target="_self">Th9</a></i>;<br/><a NAME="E8:23_1_1_3"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:23_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>;<br/><a NAME="E9:23_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E7:23_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="connsp_1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E10:23_1_1_3"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:23_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:23_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E2:23_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E4:23_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="connsp_1.html#E5:23_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="connsp_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:23_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div>
<b>assume </b><b>that </b><br/><a NAME="E2:23"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>and </b><br/><a NAME="E4:23"/><i><font color="Green">E24</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E5:23"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R1" target="_self">are_separated</a> 
 ;<br/>

<b>thus </b><a NAME="E6:23"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>4</sub></font> is <a href="connsp_1.html#V2" target="_self">connected</a> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:23"><i><font color="Green">E21</font></i></a>, <a class="txt" href="connsp_1.html#E2:23"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E3:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E4:23"><i><font color="Green">E24</font></i></a>, <a class="txt" href="connsp_1.html#E5:23"><i><font color="Green">E25</font></i></a></i>;<br/>


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