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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>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">U1</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>





<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><a NAME="E1:8_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 ;<br/><a NAME="E2:8_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">U1</font>
 <b>by </b><i>, <a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/><b>hence </b><a NAME="E3:8_1"/>
<font color="Maroon">U1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:8_2"/>
<font color="Maroon">U1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">V</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:8_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">V</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E4:8_2"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:8_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">V</font> = <font color="Maroon">V</font>;<br/><a NAME="E6:8_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E7:8_2"/>
 ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> )
 <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="E3:8"/>
( <font color="Maroon">U1</font> is    <a href="connsp_2.html#M1" target="_self">a_neighborhood</a> of <font color="Maroon">x</font> iff  ex <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">U1</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> ) )
 <b>by </b><i/>;<br/>


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