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<b>let </b><font color="Maroon">T</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">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<br/>



<b>set </b><font color="Maroon">F</font> =  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b><a NAME="E1:17"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font> implies <font color="Maroon">x</font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>,<font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:17_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5">WAYBEL_7:def 5</a><br/>

<b>assume </b><a NAME="E2:17_1"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E3:17_1"/><b>then </b>
<font color="Maroon">A</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<a NAME="E4:17_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="yellow_6.html#D18">YELLOW_6:def 18</a></i>;<br/>
<b>hence </b><a NAME="E5:17_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>
 ;<br/>


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

<b>assume </b><a NAME="E2:17"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/><font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D5">WAYBEL_7:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">O</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>;<br/><a NAME="E1:17_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/><a NAME="E2:17_2"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>
 <b>by </b><i/>;<br/><a NAME="E3:17_2"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> )
 <b>by </b><i>, <a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/><b>hence </b><a NAME="E4:17_2"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">O</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T8">WAYBEL_0:8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#D18">YELLOW_6:def 18</a></i>;<br/>


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