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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:25"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="connsp_2.html#V1" target="_self">locally_connected</a> implies for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">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">E21</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="connsp_2.html#V1" target="_self">locally_connected</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E2:25_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E3:25_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span><b> such that </b><br/><a NAME="E5:25_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:25_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="connsp_2.html#E3:25_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> ;<br/>
<a NAME="E8:25_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="connsp_2.html#V2" target="_self">locally_connected</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E1:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="connsp_2.html#E2:25_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="connsp_2.html#T23" target="_self">Th23</a></i>;<br/>
<a NAME="E9:25_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font> is <a href="connsp_2.html#V1" target="_self">locally_connected</a>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D6" target="_self">Def6</a></i>;<br/>
<a NAME="E10:25_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E6:25_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="connsp_2.html#T21" target="_self">Th21</a></i>;<br/>
<a NAME="E11:25_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:25_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E14:25_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>2</sub></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="E15:25_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E5:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="connsp_2.html#E14:25_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E17:25_1"/>
<font color="Maroon">c<sub>7</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E13:25_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E18:25_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="connsp_2.html#E2:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_2.html#E15:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tops_1.html#T38">TOPS_1:38</a></i>;<br/>


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

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_2.html#D4" target="_self">CONNSP_2:def 4</a><br/>

<a NAME="E3:25"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Olive">b<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></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">skl</a> <font color="Olive">b<sub>2</sub></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">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:25_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E2:25_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E3:25_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:25_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="connsp_1.html#K1">skl</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E5:25_2"/><i><font color="Green">E24</font></i>: 
 <a href="connsp_1.html#K1">skl</a> <font color="Maroon">c<sub>4</sub></font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>3</sub></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">c<sub>6</sub></font> =  <a href="connsp_1.html#K1">skl</a> <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="pre_topc.html#T39">PRE_TOPC:39</a></i>;<br/>
<a NAME="E7:25_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E5:25_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="connsp_1.html#D6">CONNSP_1:def 6</a></i>;<br/>
<a NAME="E8:25_2"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:25"><i><font color="Green">E21</font></i></a>, <a class="txt" href="connsp_2.html#E1:25_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E9:25_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_1.html#T40">CONNSP_1:40</a></i>;<br/>
<a NAME="E10:25_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E8:25_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="connsp_2.html#T5" target="_self">Th5</a></i>;<br/>
<a NAME="E11:25_2"/><b>then </b>
 <a href="connsp_1.html#K1">skl</a> <font color="Maroon">c<sub>4</sub></font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T12" target="_self">Th12</a></i>;<br/>
<b>hence </b><a NAME="E12:25_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></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">skl</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="connsp_2.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:25"/>
<font color="Maroon">c<sub>1</sub></font> <a href="connsp_2.html#R1" target="_self">is_locally_connected_in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="connsp_2.html#T20" target="_self">Th20</a></i>;<br/>


</div>
