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<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>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/>



<b>assume </b><a NAME="E1:23"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">X</font> <a href="connsp_2.html#R1" target="_self">is_locally_connected_in</a> <font color="Maroon">x</font>
 ;<br/>

<b>let </b><font color="Maroon">A</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>assume </b><b>that </b><br/><a NAME="E2:23"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E3:23"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</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> ;<br/>
<a NAME="E5:23"/><i><font color="Green">E25</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">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E6:23"/>
for <font color="Olive">C</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Olive">C</font> holds <br/> ex <font color="Olive">x1</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="Olive">C</font></span>)</span> st <br/>( <font color="Olive">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Olive">C</font> <a href="connsp_2.html#R1" target="_self">is_locally_connected_in</a> <font color="Olive">x1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<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 <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">C</font>
 ;<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">C</font></span>)</span> <b>by </b><i>, <a class="txt" href="connsp_2.html#E2:23"><i><font color="Green">E22</font></i></a></i>;<br/>
<b>take </b><font color="Maroon">x1</font> = <font color="Maroon">y</font>;<br/>

<a NAME="E3:23_1"/>
for <font color="Olive">U1</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>  st <font color="Olive">U1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x1</font> holds <br/> ex <font color="Olive">V</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> st <br/>( <font color="Olive">V</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x1</font> &amp; <font color="Olive">V</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">U1</font> be   <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>;<br/>

<b>assume </b><a NAME="E1:23_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">U1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x1</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">U2</font> = <font color="Maroon">U1</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="E3:23_1_1"/>
<font color="Maroon">U2</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T7" target="_self">Th7</a>, <a class="txt" href="connsp_2.html#E3:23"><i><font color="Green">E23</font></i></a>, <a class="ref" href="connsp_2.html#T8" target="_self">Th8</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E5:23_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">V</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E6:23_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">V</font> is <a href="connsp_1.html#V2">connected</a>
 <b>and </b><br/><a NAME="E7:23_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U2</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">V1</font> = <font color="Maroon">V</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="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon">V1</font>
;<br/>

<b>thus </b><a NAME="E9:23_1_1"/>
( <font color="Maroon">V1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x1</font> &amp; <font color="Maroon">V1</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">V1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> )
 <b>by </b><i><a class="txt" href="connsp_2.html#E3:23"><i><font color="Green">E23</font></i></a>, <a class="txt" href="connsp_2.html#E5:23"><i><font color="Green">E25</font></i></a>, <a class="txt" href="connsp_2.html#E1:23_1"><i><font color="Green">E26</font></i></a>, , , <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:23_1"/>
( <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">X</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">C</font> <a href="connsp_2.html#R1" target="_self">is_locally_connected_in</a> <font color="Maroon">x1</font> )
 <b>by </b><i><a class="txt" href="connsp_2.html#E3:23"><i><font color="Green">E23</font></i></a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:23"/>
<font color="Maroon">A</font> <a href="connsp_2.html#R2" target="_self">is_locally_connected_in</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D2" target="_self">Def2</a></i>;<br/>


</div>
