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<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>assume </b><a NAME="E1:24"/><i><font color="Green" title="E20">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>assume </b><a NAME="E2:24"/><i><font color="Green" title="E21">A2</font></i>: 
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E3:24"/>
for <font color="Olive" title="b1">x</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="Maroon" title="c2">A</font></span>)</span> holds  <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> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be   <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="c2">A</font></span>)</span>;<br/>

<a NAME="E1:24_1"/>
<font color="Maroon" title="c3">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="c2">A</font></span>)</span>
 
;<br/>
<a NAME="E2:24_1"/><b>then </b><i><font color="Green" title="E22">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</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>then reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E4:24_1"/>
<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="c4">x1</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E1:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="connsp_2.html#D4" target="_self" title="CONNSP_2:def.4">Def4</a></i>;<br/>
<a NAME="E5:24_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="connsp_2.html#R2" title="CONNSP_2:pred.2">is_locally_connected_in</a> <font color="Maroon" title="c4">x1</font>
 
<b>by </b><i><a class="txt" href="connsp_2.html#E2:24"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="connsp_2.html#E2:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="connsp_2.html#T22" target="_self" title="CONNSP_2:th.22">Th22</a></i>;<br/>
<a NAME="E6:24_1"/><b>then </b>
 ex <font color="Olive" title="b1">x2</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="Maroon" title="c2">A</font></span>)</span> st <br/>( <font color="Olive" title="b1">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> &amp; <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> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Olive" title="b1">x2</font> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D5" target="_self" title="CONNSP_2:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E7:24_1"/>
<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> <a href="connsp_2.html#R1" title="CONNSP_2:pred.1">is_locally_connected_in</a> <font color="Maroon" title="c3">x</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:24"/><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#D4" target="_self" title="CONNSP_2:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E5:24"/>
<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="ref" href="connsp_2.html#D6" target="_self" title="CONNSP_2:def.6">Def6</a></i>;<br/>


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