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



<a NAME="E1:21"/><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> implies for <font color="Olive">U1</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="Olive">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U1</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">U1</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="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Olive">x1</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E22</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">U1</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="E2:21_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">U1</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E3:21_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">U1</font>
 ;<br/>

<a NAME="E4:21_1"/>
<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">U1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T7" target="_self">Th7</a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</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">U1</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">x1</font>
;<br/>

<b>reconsider </b><font color="Maroon">S1</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</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">U1</font></span>)</span> ;<br/>
<a NAME="E7:21_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S1</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">U1</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">S</font> = <font color="Maroon">S1</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="E9:21_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">U1</font>
 
<b>by </b><i>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E10:21_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">U1</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#T6" target="_self">Th6</a>, <a class="ref" href="connsp_2.html#T7" target="_self">Th7</a>, </i>;<br/>
<a NAME="E11:21_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a></i>;<br/>
<a NAME="E12:21_1"/><b>then </b>
<font color="Maroon">S</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#T5" target="_self">Th5</a>, <a class="txt" href="connsp_2.html#E1:21_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_2.html#E2:21_1"><i><font color="Green">E23</font></i></a>, </i>;<br/>
<a NAME="E13:21_1"/><b>then </b>
<font color="Maroon">S1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x1</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:21_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="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</font></span>)</span> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:21"/>
( ( for <font color="Olive">U1</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="Olive">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U1</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">U1</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="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Olive">x1</font></span>)</span> ) ) implies <font color="Maroon">X</font> <a href="connsp_2.html#R1" target="_self">is_locally_connected_in</a> <font color="Maroon">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_2"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">U1</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="Olive">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U1</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">U1</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="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Olive">x1</font></span>)</span> )
 ;<br/>

<a NAME="E2:21_2"/>
for <font color="Olive">U1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">U1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font> holds <br/> ex <font color="Olive">V1</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">V1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font> &amp; <font color="Olive">V1</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Olive">V1</font> <a href="tarski.html#R1">c=</a> <font color="Olive">U1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_2_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 <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E1:21_2_1"/>
<font color="Maroon">U1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font>
 ;<br/>

<b>then consider </b><font color="Maroon">V</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><b> such that </b><br/><a NAME="E3:21_2_1"/><i><font color="Green">E36</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="E4:21_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:21_2_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font>
 <b>by </b><i/>;<br/>
<a NAME="E6:21_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6" target="_self">Th6</a>, </i>;<br/>
<b>then consider </b><font color="Maroon">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="Maroon">V</font></span>)</span><b> such that </b><br/><a NAME="E8:21_2_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E9:21_2_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</font></span>)</span>
 <b>by </b><i><a class="ref" href="connsp_2.html#T5" target="_self">Th5</a>, <a class="ref" href="connsp_2.html#T7" target="_self">Th7</a></i>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</font>;<br/>
<a NAME="E10:21_2_1"/><i><font color="Green">E59</font></i>: 
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x1</font>
 
<b>by </b><i>, , </i>;<br/>
<b>reconsider </b><font color="Maroon">S</font> =  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</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>
;<br/>

<a NAME="E12:21_2_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">X</font> &amp;  <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</font> is <a href="connsp_1.html#V2">connected</a> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a>, <a class="ref" href="connsp_1.html#T41">CONNSP_1:41</a></i>;<br/>
<a NAME="E13:21_2_1"/>
 <a href="connsp_1.html#K1">Component_of</a> <font color="Maroon">x1</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">V</font></span>)</span>
 
;<br/>
<a NAME="E14:21_2_1"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>hence </b><a NAME="E15:21_2_1"/>
( <font color="Maroon">S</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font> &amp; <font color="Maroon">S</font> is <a href="connsp_1.html#V2">connected</a> &amp; <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> )
 <b>by </b><i><a class="ref" href="connsp_2.html#T7" target="_self">Th7</a>, , , <a class="ref" href="connsp_2.html#T8" target="_self">Th8</a>, , , <a class="ref" href="connsp_1.html#T24">CONNSP_1:24</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_2"/>
<font color="Maroon">X</font> <a href="connsp_2.html#R1" target="_self">is_locally_connected_in</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
( <font color="Maroon">X</font> <a href="connsp_2.html#R1" target="_self">is_locally_connected_in</a> <font color="Maroon">x</font> iff for <font color="Olive">U1</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="Olive">U1</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U1</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">U1</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="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="connsp_1.html#K1">Component_of</a> <font color="Olive">x1</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


</div>
