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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:28"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">X</font> is <a href="connsp_2.html#V1" target="_self">locally_connected</a> &amp; <font color="Maroon">X</font> <a href="compts_1.html#NR4" target="_self">is_T4</a>  )
 ;<br/>

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

<b>assume </b><b>that </b><br/><a NAME="E2:28"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>and </b><br/><a NAME="E3:28"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">B</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>and </b><br/><a NAME="E4:28"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 ;<br/>

<b>assume </b><a NAME="E5:28"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a> )
 ;<br/>

<a NAME="E6:28"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="connsp_2.html#E4:28"><i><font color="Green">E25</font></i></a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E7:28"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T8" target="_self">Th8</a>, <a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E8:28"/>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</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#K7">\</a> <font color="Maroon">B</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T22">PRE_TOPC:22</a></i>;<br/>
<a NAME="E9:28"/><b>then </b>
<span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="subset_1.html#K7">\</a> <font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#T23">PRE_TOPC:23</a></i>;<br/>
<a NAME="E10:28"/><b>then </b>
<font color="Maroon">B</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
;<br/>
<b>then consider </b><font color="Maroon">G</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E12:28"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> )
 <b>and </b><br/><a NAME="E13:28"/><i><font color="Green">E36</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="connsp_2.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_2.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="ref" href="connsp_2.html#T8" target="_self">Th8</a>, , <a class="txt" href="connsp_2.html#E6:28"><i><font color="Green">E28</font></i></a>, </i>;<br/>
<a NAME="E14:28"/><i><font color="Green">E55</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E15:28"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E3:28"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E17:28"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T41">PRE_TOPC:41</a></i>;<br/>
<a NAME="E18:28"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E7:28"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="txt" href="connsp_2.html#E3:28"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_2.html#E7:28"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E20:28"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <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">G</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">G</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">H</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font>;<br/>
<b>reconsider </b><font color="Maroon">H1</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/>
<b>take </b><font color="Maroon">R</font> = <font color="Maroon">H1</font>;<br/>

<a NAME="E23:28"/><i><font color="Green">E58</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T43">CONNSP_1:43</a></i>;<br/>
<a NAME="E24:28"/><i><font color="Green">E59</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T41">CONNSP_1:41</a></i>;<br/>
<a NAME="E25:28"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">H1</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E26:28"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <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">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E7:28"><i><font color="Green">E29</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</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">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">G</font></span>)</span> ;<br/>
<a NAME="E28:28"/>
<font color="Maroon">A1</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E5:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E29:28"/><b>then </b>
( <font color="Maroon">A1</font> <a href="xboole_0.html#R1">misses</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font> or <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/>
<a NAME="E30:28"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">A1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">A1</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E31:28"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a></i>;<br/>
<a NAME="E32:28"/>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E3:28"><i><font color="Green">E23</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">z</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E34:28"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T41">PRE_TOPC:41</a></i>;<br/>
<a NAME="E35:28"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="connsp_2.html#T11" target="_self">Th11</a>, <a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E36:28"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">C</font> = <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i>, <a class="ref" href="connsp_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E38:28"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <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">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T13" target="_self">Th13</a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">z1</font> = <font color="Maroon">z</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">V</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/>
<b>take </b><font color="Maroon">S</font> = <font color="Maroon">V1</font>;<br/>

<a NAME="E41:28"/><i><font color="Green">E82</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T43">CONNSP_1:43</a></i>;<br/>
<a NAME="E42:28"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/>
<a NAME="E43:28"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</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/>
<a NAME="E44:28"/><b>then </b><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>
<a NAME="E45:28"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">V1</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E46:28"/><i><font color="Green">E88</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T41">CONNSP_1:41</a></i>;<br/>
<a NAME="E47:28"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T12" target="_self">Th12</a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then 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">X</font> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span> ;<br/>
<a NAME="E49:28"/>
<font color="Maroon">B1</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E5:28"><i><font color="Green">E26</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E50:28"/><b>then </b>
( <font color="Maroon">B1</font> <a href="xboole_0.html#R1">misses</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font> or <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font> )
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/>
<a NAME="E51:28"/><b>then </b><i><font color="Green">E89</font></i>: 
( <font color="Maroon">B1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B1</font> <a href="tarski.html#R1">c=</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E52:28"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">z1</font> <a href="hidden.html#R2">in</a>  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a></i>;<br/>
<a NAME="E53:28"/>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <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">G</font></span>)</span>
 
;<br/>
<a NAME="E54:28"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E55:28"/>
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E56:28"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">R</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D5" target="_self">Def5</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E57:28"/>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">z1</font> <a href="tarski.html#R1">c=</a>  <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> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span></span>)</span>
 
;<br/>
<a NAME="E58:28"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E59:28"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">R</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D6" target="_self">Def6</a>, <a class="ref" href="xboole_1.html#T27">XBOOLE_1:27</a></i>;<br/>
<a NAME="E60:28"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E61:28"/><b>then </b>
<font color="Maroon">R</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T19" target="_self">Th19</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E62:28"/><b>then </b>
<font color="Maroon">R</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<b>hence </b><a NAME="E63:28"/>
( <font color="Maroon">R</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">S</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">R</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">R</font> &amp; <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> &amp; <font color="Maroon">R</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">S</font> )
 <b>by </b><i><a class="txt" href="connsp_2.html#E2:28"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_2.html#E7:28"><i><font color="Green">E29</font></i></a>, , <a class="txt" href="connsp_2.html#E12:28"><i><font color="Green">E35</font></i></a>, <a class="txt" href="connsp_2.html#E13:28"><i><font color="Green">E36</font></i></a>, , <a class="txt" href="connsp_2.html#E14"><i><font color="Green">Lemma38</font></i></a>, , , , , <a class="ref" href="connsp_2.html#D3" target="_self">Def3</a>, <a class="ref" href="connsp_2.html#D4" target="_self">Def4</a>, <a class="ref" href="connsp_2.html#T7" target="_self">Th7</a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>


</div>
