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<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>assume </b><a NAME="E1:22"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">X</font> is <a href="connsp_2.html#V1" target="_self">locally_connected</a>
 ;<br/>

<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:22"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">S</font> <a href="connsp_1.html#R3">is_a_component_of</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:22"/><i><font color="Green">E23</font></i>: 
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/><b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> ;<br/><a NAME="E3:22_1"/><i><font color="Green">E26</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>
 <b>by </b><i><a class="ref" href="connsp_2.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E4:22_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S</font> <a href="connsp_1.html#R4">is_a_component_of</a>  <a href="pre_topc.html#K2">[#]</a> <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#D1" target="_self">Def1</a></i>;<br/><a NAME="E5:22_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> &amp;  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/><a NAME="E6:22_1"/><b>then </b>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font>
 <b>by </b><i/>;<br/><a NAME="E7:22_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="txt" href="connsp_2.html#E2:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="connsp_2.html#E3:22"><i><font color="Green">E23</font></i></a>, </i>;<br/><b>hence </b><a NAME="E8:22_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E5:22"/><b>then </b>
<font color="Maroon">S</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E6:22"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T7" target="_self">Th7</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:22"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>


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