<?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="E1:26"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">X</font> is <a href="connsp_2.html#V1" target="_self">locally_connected</a>
 ;<br/>

<b>let </b><font color="Maroon">E</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a> <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">E</font></span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:26"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">C</font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E3:26"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">C</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E4:26"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">E</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> &amp; <font color="Maroon">C</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">E</font></span>)</span> )
 
;<br/>
<a NAME="E5:26"/>
<font color="Maroon">C</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">E</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E3:26"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<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="E7:26"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">G</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E8:26"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</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">E</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E9:26"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E7:26"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</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#E7:26"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being   <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">E</font></span>)</span><b> such that </b><br/><a NAME="E12:26"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T8" target="_self">Th8</a>, <a class="ref" href="pre_topc.html#T41">PRE_TOPC:41</a></i>;<br/>
<a NAME="E13:26"/>
<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#E7:26"><i><font color="Green">E26</font></i></a>, <a class="txt" href="connsp_2.html#E8:26"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:26"/><b>then </b>
<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">H1</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">y</font>;<br/>
<a NAME="E16:26"/><i><font color="Green">E36</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/>
<b>reconsider </b><font color="Maroon">H</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/>
<a NAME="E18:26"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">H</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E9:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<a NAME="E19:26"/>
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E4:26"><i><font color="Green">E25</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E20:26"/>
<font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="connsp_2.html#T7" target="_self">Th7</a></i>;<br/>

<a NAME="E21:26"/>
 <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="txt" href="connsp_2.html#E9:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="connsp_1.html#D5">CONNSP_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E22:26"/>
<font color="Maroon">H</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>

<b>reconsider </b><font color="Maroon">C1</font> = <font color="Maroon">C</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/>
<a NAME="E24:26"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">C1</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:26"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E25:26"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E7:26"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E26:26"/><b>then </b>
<font color="Maroon">C</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="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C2</font> = <font color="Maroon">C1</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="E28:26"/>
<font color="Maroon">C2</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T11" target="_self">Th11</a>, <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>
<a NAME="E29:26"/><b>then </b>
( <font color="Maroon">C2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">H</font> or <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="txt" href="connsp_2.html#E9:26"><i><font color="Green">E29</font></i></a>, <a class="ref" href="connsp_1.html#T38">CONNSP_1:38</a></i>;<br/>
<a NAME="E30:26"/><b>then </b><i><font color="Green">E57</font></i>: 
( <font color="Maroon">C2</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H</font> <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">G</font></span>)</span> or <font color="Maroon">C2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">H</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E31:26"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</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="E32:26"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E33:26"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E8:26"><i><font color="Green">E28</font></i></a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E34:26"/>
<font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="txt" href="connsp_2.html#E8:26"><i><font color="Green">E28</font></i></a>, , , <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E35:26"/><b>then </b><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">G</font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T16">XBOOLE_1:16</a></i>;<br/>
<a NAME="E36:26"/>
 <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="E37:26"/><b>then </b>
<font color="Maroon">H</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="E38:26"/><b>then </b>
<font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E12:26"><i><font color="Green">E35</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E39:26"/>
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">H</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
