<?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>thus </b><a NAME="E1:25"/>
( <font color="Maroon" title="c1">X</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a> implies for <font color="Olive" title="b1">A</font> being 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><br/> for <font color="Olive" title="b2">B</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">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c1">X</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c2">A</font> be  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>;<br/><b>let </b><font color="Maroon" title="c3">B</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="E2:25_1"/><i><font color="Green" title="E22">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E3:25_1"/><i><font color="Green" title="E23">A3</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c2">A</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">B1</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="c2">A</font></span>)</span><b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c4">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">B</font>
 <b>and </b><br/><a NAME="E6:25_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c4">B1</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="c2">A</font>
 <b>by </b><i><a class="txt" href="connsp_2.html#E3:25_1"><i><font color="Green" title="E23">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="c5">B1</font> = <font color="Maroon" title="c4">B1</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="c2">A</font></span>)</span> ;<br/>
<a NAME="E8:25_1"/>
<font color="Maroon" title="c2">A</font> is <a href="connsp_2.html#V2" title="CONNSP_2:attr.2">locally_connected</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E1:25_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="connsp_2.html#E2:25_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="connsp_2.html#T23" target="_self" title="CONNSP_2:th.23">Th23</a></i>;<br/>
<a NAME="E9:25_1"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">A</font> is <a href="connsp_2.html#V1" title="CONNSP_2:attr.1">locally_connected</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D6" target="_self" title="CONNSP_2:def.6">Def6</a></i>;<br/>
<a NAME="E10:25_1"/><b>then </b>
<font color="Maroon" title="c5">B1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E6:25_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="connsp_2.html#T21" target="_self" title="CONNSP_2:th.21">Th21</a></i>;<br/>
<a NAME="E11:25_1"/><b>then </b>
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</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="c2">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">B2</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="E13:25_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c6">B2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font>
 <b>and </b><br/><a NAME="E14:25_1"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c5">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<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="c2">A</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E15:25_1"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">B2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">A</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E5:25_1"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="connsp_2.html#E14:25_1"><i><font color="Green" title="E27">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>reconsider </b><font color="Maroon" title="c7">B2</font> = <font color="Maroon" title="c6">B2</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E17:25_1"/>
<font color="Maroon" title="c7">B2</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E13:25_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E18:25_1"/>
<font color="Maroon" title="c3">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="connsp_2.html#E2:25_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="connsp_2.html#E15:25_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="tops_1.html#T38" title="TOPS_1:th.38">TOPS_1:38</a></i>;<br/>


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

<b>assume </b><a NAME="E2:25"/><i><font color="Green" title="E21">A9</font></i>: 
for <font color="Olive" title="b1">A</font> being 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><br/> for <font color="Olive" title="b2">B</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">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b2">B</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Olive" title="b1">A</font> holds <br/><font color="Olive" title="b2">B</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D4" target="_self" title="CONNSP_2:def.4">CONNSP_2:def 4</a><br/>

<a NAME="E3:25"/>
for <font color="Olive" title="b1">U1</font> being 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>  st <font color="Olive" title="b1">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">U1</font> holds <br/> ex <font color="Olive" title="b2">x1</font> being  <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="Olive" title="b1">U1</font></span>)</span> st <br/>( <font color="Olive" title="b2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <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> <span class="p1">(<span class="default"><a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Olive" title="b2">x1</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">U1</font> be  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>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_2"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c3">U1</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E2:25_2"/><i><font color="Green" title="E23">A11</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>
 ;<br/>

<a NAME="E3:25_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="c3">U1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:25_2"><i><font color="Green" title="E23">A11</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="c4">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="c3">U1</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c4">x1</font>
;<br/>

<b>set </b><font color="Maroon" title="c5">S1</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c4">x1</font>;<br/>
<a NAME="E5:25_2"/><i><font color="Green" title="E24">A12</font></i>: 
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c4">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="c3">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="c6">S</font> =  <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c4">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/>
<a NAME="E7:25_2"/>
<font color="Maroon" title="c6">S</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c3">U1</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E5:25_2"><i><font color="Green" title="E24">A12</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="E8:25_2"/><b>then </b><i><font color="Green" title="E25">A13</font></i>: 
<font color="Maroon" title="c6">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:25"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="connsp_2.html#E1:25_2"><i><font color="Green" title="E22">A10</font></i></a></i>;<br/>
<a NAME="E9:25_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">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="E10:25_2"/><b>then </b>
<font color="Maroon" title="c6">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#E8:25_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="connsp_2.html#T5" target="_self" title="CONNSP_2:th.5">Th5</a></i>;<br/>
<a NAME="E11:25_2"/><b>then </b>
 <a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c4">x1</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c4">x1</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T12" target="_self" title="CONNSP_2:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E12:25_2"/>
( <font color="Maroon" title="c4">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <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> <span class="p1">(<span class="default"><a href="connsp_1.html#K1" title="CONNSP_1:func.1">Component_of</a> <font color="Maroon" title="c4">x1</font></span>)</span> )
 <b>by </b><i><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>hence </b><a NAME="E4:25"/>
<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#T20" target="_self" title="CONNSP_2:th.20">Th20</a></i>;<br/>


</div>
