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

<b>let </b><font color="Maroon">GX</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">X'</font> be    <a href="pre_topc.html#M1">SubSpace</a> of <font color="Maroon">GX</font>;<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">X'</font>;<br/>







<b>assume </b><a NAME="E1:26"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 ;<br/>

<a NAME="E2:26"/><i><font color="Green">E24</font></i>: 
(  <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> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">B</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">GX'</font> = <font color="Maroon">GX</font>, <font color="Maroon">X</font> = <font color="Maroon">X'</font> as   <a href="pre_topc.html#NM1">TopSpace</a> ;<br/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX'</font> ;<br/>
<b>reconsider </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E6:26"/><i><font color="Green">E25</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E7:26"/><i><font color="Green">E27</font></i>: 
 <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
 
;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/>
not <font color="Maroon">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 ;<br/><a NAME="E2:26_1"/><b>then </b>
not <font color="Maroon">GX'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A'</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">P</font> being   <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> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">Q</font> being   <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> <font color="Maroon">A</font></span>)</span><b> such that </b><br/><a NAME="E4:26_1"/><i><font color="Green">E30</font></i>: 
 <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> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E5:26_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E6:26_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>
 <b>and </b><br/><a NAME="E7:26_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E8:26_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E9:26_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E11:26_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E12:26_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">P1</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> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E4:26_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">Q1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E14:26_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">Q1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E15:26_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Q1</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> <font color="Maroon">A</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E5:26_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>reconsider </b><font color="Maroon">PP</font> = <font color="Maroon">P</font>, <font color="Maroon">QQ</font> = <font color="Maroon">Q</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">P11</font> = <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font> ;<br/><a NAME="E18:26_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">P11</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E7:26_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E19:26_1"/>
( <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E20:26_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="connsp_1.html#T6" target="_self">Th6</a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E21:26_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E22:26_1"/>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E23:26_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="connsp_1.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E24:26_1"/><b>then </b>
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E25:26_1"/><b>then </b>
<font color="Maroon">P11</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">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">PP</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E26:26_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">PP</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>reconsider </b><font color="Maroon">Q11</font> = <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font> ;<br/><a NAME="E28:26_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">Q11</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E29:26_1"/>
( <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E30:26_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, , <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/><a NAME="E31:26_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E32:26_1"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E33:26_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span>
 <b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, , <a class="ref" href="connsp_1.html#T8" target="_self">Th8</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E34:26_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/><a NAME="E35:26_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">Q1</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="ref" href="connsp_1.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><a NAME="E36:26_1"/><b>then </b>
<font color="Maroon">QQ</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, , <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E37:26_1"/><b>then </b>
not <font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B'</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><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#E6:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E7:26"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a>, <a class="txt" href="connsp_1.html#E8:26"><i><font color="Green">E29</font></i></a>, <a class="txt" href="connsp_1.html#E6:26_1"><i><font color="Green">E32</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E38:26_1"/>
not <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/>
not <font color="Maroon">B</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 ;<br/><a NAME="E2:26_2"/><b>then </b>
not <font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span><b> such that </b><br/><a NAME="E4:26_2"/><i><font color="Green">E80</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Q</font>
 <b>and </b><br/><a NAME="E5:26_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
 <b>and </b><br/><a NAME="E6:26_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span>
 <b>and </b><br/><a NAME="E7:26_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">P</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E8:26_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">Q</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E9:26_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">P</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">Q</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">P1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font><b> such that </b><br/><a NAME="E11:26_2"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">P1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E12:26_2"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">P1</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">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">Q1</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X'</font><b> such that </b><br/><a NAME="E14:26_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">Q1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E15:26_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">Q1</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">X'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">B</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">P2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E17:26_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">P2</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E18:26_2"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>consider </b><font color="Maroon">Q2</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font><b> such that </b><br/><a NAME="E20:26_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">Q2</font> is <a href="pre_topc.html#V4">closed</a>
 <b>and </b><br/><a NAME="E21:26_2"/><i><font color="Green">E93</font></i>: 
<font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q1</font>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><b>reconsider </b><font color="Maroon">PP</font> = <font color="Maroon">P</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> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">QQ</font> = <font color="Maroon">Q</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> <font color="Maroon">A</font></span>)</span> <b>by </b><i><a class="txt" href="connsp_1.html#E2:26"><i><font color="Green">E24</font></i></a></i>;<br/><font color="Maroon">P2</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> <font color="Maroon">A</font></span>)</span></span>)</span> = 
<font color="Maroon">P2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">PP</font>
<b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="connsp_1.html#T6" target="_self">Th6</a>, , <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/><a NAME="E25:26_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">PP</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><font color="Maroon">Q2</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> <font color="Maroon">A</font></span>)</span></span>)</span> = 
<font color="Maroon">Q2</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X'</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">QQ</font>
<b>by </b><i><a class="ref" href="connsp_1.html#T3" target="_self">Th3</a>, , , <a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>
;<br/><a NAME="E27:26_2"/><b>then </b>
<font color="Maroon">QQ</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T43">PRE_TOPC:43</a></i>;<br/><a NAME="E28:26_2"/><b>then </b>
not <font color="Maroon">GX'</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A'</font> is <a href="connsp_1.html#V1" target="_self">connected</a>
 <b>by </b><i><a class="txt" href="connsp_1.html#E1:26"><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#E6:26"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_1.html#E7:26"><i><font color="Green">E27</font></i></a>, <a class="ref" href="connsp_1.html#T5" target="_self">Th5</a>, <a class="txt" href="connsp_1.html#E8:26"><i><font color="Green">E29</font></i></a>, , , </i>;<br/><b>hence </b><a NAME="E29:26_2"/>
not <font color="Maroon">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:26"/>
( <font color="Maroon">A</font> is <a href="connsp_1.html#V2" target="_self">connected</a> iff <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#T4" target="_self">Th4</a></i>;<br/>


</div>
