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

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/>



<b>thus </b><a NAME="E1:20"/>
( <font color="Maroon" title="c1">X</font> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Maroon" title="c2">x</font> implies for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">S</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> holds <br/><font color="Olive" title="b2">S</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:20_1"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Maroon" title="c2">x</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">V</font>, <font color="Maroon" title="c4">S</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E2:20_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c3">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E3:20_1"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c4">S</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c3">V</font>
 <b>and </b><br/><a NAME="E4:20_1"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">V1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E6:20_1"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c5">V1</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>
 <b>and </b><br/><a NAME="E7:20_1"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c5">V1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>and </b><br/><a NAME="E8:20_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c5">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">V</font>
 <b>by </b><i><a class="txt" href="connsp_2.html#E1:20_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="connsp_2.html#E2:20_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="connsp_2.html#D3" target="_self" title="CONNSP_2:def.3">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">S1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">V</font></span>)</span><b> such that </b><br/><a NAME="E10:20_1"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon" title="c6">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">S</font> &amp; <font color="Maroon" title="c6">S1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">V</font> )
 <b>by </b><i><a class="txt" href="connsp_2.html#E3:20_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">V'</font> = <font color="Maroon" title="c3">V</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="connsp_2.html#E2:20_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="connsp_2.html#T6" target="_self" title="CONNSP_2:th.6">Th6</a></i>;<br/>
<a NAME="E12:20_1"/>
<font color="Maroon" title="c5">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">V</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E8:20_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">V2</font> = <font color="Maroon" title="c5">V1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">V</font></span>)</span> ;<br/>
<a NAME="E14:20_1"/>
<font color="Maroon" title="c8">V2</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E7:20_1"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>
<a NAME="E15:20_1"/><b>then </b>
( <font color="Maroon" title="c8">V2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">S1</font> or <font color="Maroon" title="c8">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">S1</font> )
 
<b>by </b><i><a class="txt" href="connsp_2.html#E10:20_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="connsp_1.html#T38" title="CONNSP_1:th.38">CONNSP_1:38</a></i>;<br/>
<a NAME="E16:20_1"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
( <font color="Maroon" title="c8">V2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c6">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">V'</font></span>)</span> or <font color="Maroon" title="c8">V2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">S1</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E17:20_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V2</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">S1</font> )
 
<b>by </b><i><a class="txt" href="connsp_2.html#E4:20_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="connsp_2.html#E6:20_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="connsp_2.html#E10:20_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="connsp_2.html#T6" target="_self" title="CONNSP_2:th.6">Th6</a></i>;<br/>
<a NAME="E18:20_1"/><b>then </b><i><font color="Green" title="E25">A10</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c4">S</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E10:20_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="connsp_2.html#E16:20_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="tops_1.html#T48" title="TOPS_1:th.48">TOPS_1:48</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E19:20_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c5">V1</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E6:20_1"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="connsp_2.html#D1" target="_self" title="CONNSP_2:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E20:20_1"/>
<font color="Maroon" title="c4">S</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="txt" href="connsp_2.html#E18:20_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="connsp_2.html#D1" target="_self" title="CONNSP_2:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:20"/><i><font color="Green" title="E16">A11</font></i>: 
for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">S</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">S</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">S</font> holds <br/><font color="Olive" title="b2">S</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E3:20"/>
for <font color="Olive" title="b1">U1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">U1</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> holds <br/> ex <font color="Olive" title="b2">V</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b2">V</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">V</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">U1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">U1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green" title="E17">A12</font></i>: 
<font color="Maroon" title="c3">U1</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font>
 ;<br/>

<a NAME="E2:20_2"/><b>then </b><i><font color="Green" title="E18">A13</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">U1</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T6" target="_self" title="CONNSP_2:th.6">Th6</a></i>;<br/>
<a NAME="E3:20_2"/><i><font color="Green" title="E19">A14</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">U1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">U1</font> = <font color="Maroon" title="c3">U1</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="txt" href="connsp_2.html#E1:20_2"><i><font color="Green" title="E17">A12</font></i></a>, <a class="ref" href="connsp_2.html#T6" target="_self" title="CONNSP_2:th.6">Th6</a></i>;<br/>
<a NAME="E5:20_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:20_2"><i><font color="Green" title="E18">A13</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c2">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U1</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">S1</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c5">x1</font>;<br/>
<a NAME="E7:20_2"/><i><font color="Green" title="E20">A15</font></i>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c5">x1</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c4">U1</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T43" title="CONNSP_1:th.43">CONNSP_1:43</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">S</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c5">x1</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> <b>by </b><i><a class="ref" href="pre_topc.html#T39" title="PRE_TOPC:th.39">PRE_TOPC:39</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">S</font>
;<br/>

<a NAME="E9:20_2"/><i><font color="Green" title="E21">A16</font></i>: 
<font color="Maroon" title="c7">S</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c4">U1</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E7:20_2"><i><font color="Green" title="E20">A15</font></i></a>, <a class="ref" href="connsp_1.html#D6" title="CONNSP_1:def.6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E10:20_2"/><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40" title="CONNSP_1:th.40">CONNSP_1:40</a></i>;<br/>
<a NAME="E11:20_2"/>
( <font color="Maroon" title="c7">S</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c5">x1</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40" title="CONNSP_1:th.40">CONNSP_1:40</a>, <a class="ref" href="connsp_1.html#T41" title="CONNSP_1:th.41">CONNSP_1:41</a></i>;<br/>
<b>hence </b><a NAME="E12:20_2"/>
( <font color="Maroon" title="c7">S</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c7">S</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp; <font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">U1</font> )
 <b>by </b><i><a class="txt" href="connsp_2.html#E2:20"><i><font color="Green" title="E16">A11</font></i></a>, <a class="txt" href="connsp_2.html#E1:20_2"><i><font color="Green" title="E17">A12</font></i></a>, <a class="txt" href="connsp_2.html#E3:20_2"><i><font color="Green" title="E19">A14</font></i></a>, <a class="txt" href="connsp_2.html#E9:20_2"><i><font color="Green" title="E21">A16</font></i></a>, <a class="txt" href="connsp_2.html#E10:20_2"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="connsp_1.html#T24" title="CONNSP_1:th.24">CONNSP_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:20"/>
<font color="Maroon" title="c1">X</font> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D3" target="_self" title="CONNSP_2:def.3">Def3</a></i>;<br/>


</div>
