<?xml version="1.0"?>
<div class="add">

<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="24_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="E1:24_1"/><i><font color="Green">E22</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="E2:24_1"/><i><font color="Green">E23</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> 
 <b>and </b><br/><a NAME="E3:24_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E4:24_1"/><i><font color="Green">E25</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E3:24_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/><a NAME="E5:24_1"/><i><font color="Green">E26</font></i>: 
 <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>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#E1:24_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E6:24_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/><a NAME="E7:24_1"/>
(  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#R1">misses</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E2:24_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="connsp_1.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E8:24_1"/><b>then </b><i><font color="Green">E28</font></i>: 
( <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/> <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#T50">PRE_TOPC:50</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<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></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E4:24_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E5:24_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p3">(<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></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T24">XBOOLE_1:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T4">XBOOLE_1:4</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<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></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T24">XBOOLE_1:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="connsp_1.html#E6:24_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="connsp_1.html#E8:24_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>hence </b><a NAME="E10:24_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="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/></div><b>end;</b></div>
<b>assume </b><b>that </b><br/><a NAME="E2:24"/><i><font color="Green">E22</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="E3:24"/><i><font color="Green">E23</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> 
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>thus </b><a NAME="E5:24"/>
( <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="pre_topc.html#V4">closed</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="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:24"><i><font color="Green">E21</font></i></a>, <a class="txt" href="connsp_1.html#E2:24"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_1.html#E3:24"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_1.html#E4:24"><i><font color="Green">E24</font></i></a></i>;<br/>


</div>
