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

<b>let </b><font color="Maroon" title="c1">GX</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">B</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">GX</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:27"/><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E2:27"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:27"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E4:27"/><i><font color="Green" title="E25">A4</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c3">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 ;<br/>

<a NAME="E5:27"/><i><font color="Green" title="E26">A5</font></i>: 
<font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span> is <a href="connsp_1.html#V1" title="CONNSP_1:attr.1">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E3:27"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="connsp_1.html#D3" target="_self" title="CONNSP_1:def.3">Def3</a></i>;<br/>
<a NAME="E6:27"/><i><font color="Green" title="E27">A6</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">AB</font> = <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font>;<br/>
<a NAME="E7:27"/><i><font color="Green" title="E28">A7</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> &amp; <font color="Maroon" title="c3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">A1</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E6:27"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">B1</font> = <font color="Maroon" title="c3">B</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E6:27"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="connsp_1.html#E7:27"><i><font color="Green" title="E28">A7</font></i></a></i>;<br/>
<a NAME="E10:27"/><i><font color="Green" title="E29">A8</font></i>: 
<span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">B1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E6:27"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="xboole_1.html#T55" title="XBOOLE_1:th.55">XBOOLE_1:55</a></i>;<br/>
<a NAME="E11:27"/><i><font color="Green" title="E30">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">B1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E12:27"/><i><font color="Green" title="E31">A10</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T51" title="XBOOLE_1:th.51">XBOOLE_1:51</a></i>;<br/>
<a NAME="E13:27"/><i><font color="Green" title="E32">A11</font></i>: 
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E6:27"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="connsp_1.html#E7:27"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E14:27"/>
<font color="Maroon" title="c3">B</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">GX</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">B</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E6:27"><i><font color="Green" title="E27">A6</font></i></a>, <a class="txt" href="connsp_1.html#E7:27"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E15:27"/><b>then </b>
( <font color="Maroon" title="c5">A1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c6">B1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="connsp_1.html#E1:27"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="connsp_1.html#E2:27"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="connsp_1.html#E13:27"><i><font color="Green" title="E32">A11</font></i></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E16:27"/><b>then </b><i><font color="Green" title="E33">A12</font></i>: 
<font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c6">B1</font>,<font color="Maroon" title="c6">B1</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">A1</font> <a href="connsp_1.html#R1" title="CONNSP_1:pred.1">are_separated</a> 
 
<b>by </b><i><a class="ref" href="connsp_1.html#T10" target="_self" title="CONNSP_1:th.10">Th10</a></i>;<br/>
<a NAME="E17:27"/>
<font color="Maroon" title="c5">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">B1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E4:27"><i><font color="Green" title="E25">A4</font></i></a>, <a class="ref" href="connsp_1.html#T24" target="_self" title="CONNSP_1:th.24">Th24</a></i>;<br/>
<a NAME="E18:27"/><b>then </b>
( <font color="Maroon" title="c5">A1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c6">B1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 
<b>by </b><i><a class="txt" href="connsp_1.html#E5:27"><i><font color="Green" title="E26">A5</font></i></a>, <a class="txt" href="connsp_1.html#E10:27"><i><font color="Green" title="E29">A8</font></i></a>, <a class="txt" href="connsp_1.html#E11:27"><i><font color="Green" title="E30">A9</font></i></a>, <a class="txt" href="connsp_1.html#E12:27"><i><font color="Green" title="E31">A10</font></i></a>, <a class="txt" href="connsp_1.html#E16:27"><i><font color="Green" title="E33">A12</font></i></a>, <a class="ref" href="connsp_1.html#T21" target="_self" title="CONNSP_1:th.21">Th21</a></i>;<br/>
<b>hence </b><a NAME="E19:27"/>
( <font color="Maroon" title="c2">A</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c3">B</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 <b>by </b><i><a class="ref" href="connsp_1.html#T24" target="_self" title="CONNSP_1:th.24">Th24</a></i>;<br/>


</div>
