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<b>let </b><font color="Maroon" title="c1">T</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>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><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 <font color="Maroon" title="c1">T</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c4">F</font> =  <a href="yellow19.html#K1" title="YELLOW19:func.1">NeighborhoodSystem</a> <font color="Maroon" title="c3">x</font> as  <a href="tex_2.html#V1" title="TEX_2:attr.1">proper</a> <a href="waybel_0.html#NM4" title="WAYBEL_0:NM.4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font></span>)</span></span>)</span> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c5">N</font> = <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="yellow_6.html#K6" title="YELLOW_6:func.6">"</a> <font color="Maroon" title="c2">A</font> as    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of  <a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font> <b>by </b><i><a class="ref" href="yellow19.html#T20" target="_self" title="YELLOW19:th.20">Th20</a>, <a class="ref" href="yellow_6.html#T31" title="YELLOW_6:th.31">YELLOW_6:31</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">N'</font> = <font color="Maroon" title="c5">N</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font> ;<br/><b>take </b><font color="Maroon" title="c7">N'</font> = <font color="Maroon" title="c6">N'</font>;<br/><b>thus </b><a NAME="E4:29_1"/>
<font color="Maroon" title="c7">N'</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="yellow_6.html#T32" title="YELLOW_6:th.32">YELLOW_6:32</a></i>;<br/><a NAME="E5:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="waybel_7.html#R2" title="WAYBEL_7:pred.2">is_a_convergence_point_of</a> <font color="Maroon" title="c4">F</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="ref" href="yellow19.html#T4" target="_self" title="YELLOW19:th.4">Th4</a></i>;<br/><a NAME="E6:29_1"/><b>then </b>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span> &amp;  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c4">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="yellow_6.html#K11" title="YELLOW_6:func.11">Lim</a> <font color="Maroon" title="c5">N</font> )
 <b>by </b><i><a class="ref" href="yellow19.html#T18" target="_self" title="YELLOW19:th.18">Th18</a>, <a class="ref" href="yellow_6.html#T41" title="YELLOW_6:th.41">YELLOW_6:41</a></i>;<br/><b>hence </b><a NAME="E7:29_1"/>
<font color="Maroon" title="c3">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c7">N'</font>
 <b>by </b><i><a class="ref" href="waybel_9.html#T29" title="WAYBEL_9:th.29">WAYBEL_9:29</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c5">N</font> being   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E4:29"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c5">N</font> <a href="waybel_0.html#R1" title="WAYBEL_0:pred.1">is_eventually_in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">x</font> <a href="waybel_9.html#R3" title="WAYBEL_9:pred.3">is_a_cluster_point_of</a> <font color="Maroon" title="c5">N</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c6">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">N</font><b> such that </b><br/><a NAME="E6:29"/><i><font color="Green" title="E21">A2</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">N</font>  st <font color="Maroon" title="c6">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c5">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Olive" title="b1">j</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="txt" href="yellow19.html#E4:29"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel_0.html#D11" title="WAYBEL_0:def.11">WAYBEL_0:def 11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><a NAME="E1:29_2"/><i><font color="Green" title="E22">A3</font></i>: 
( <font color="Maroon" title="c7">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> )
 ;<br/><a NAME="E2:29_2"/><b>then </b>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <font color="Maroon" title="c7">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font>
 <b>by </b><i><a class="ref" href="tops_1.html#T55" title="TOPS_1:th.55">TOPS_1:55</a></i>;<br/><a NAME="E3:29_2"/><b>then </b>
<font color="Maroon" title="c7">G</font> is    <a href="connsp_2.html#M1" title="CONNSP_2:mode.1">a_neighborhood</a> of <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E1:29_2"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="connsp_2.html#D1" title="CONNSP_2:def.1">CONNSP_2:def 1</a></i>;<br/><a NAME="E4:29_2"/><b>then </b>
<font color="Maroon" title="c5">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c7">G</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E4:29"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="waybel_9.html#D9" title="WAYBEL_9:def.9">WAYBEL_9:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c5">N</font><b> such that </b><br/><a NAME="E6:29_2"/><i><font color="Green" title="E23">A4</font></i>: 
( <font color="Maroon" title="c6">i</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c5">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">G</font> )
 <b>by </b><i><a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></i>;<br/><a NAME="E7:29_2"/>
<font color="Maroon" title="c5">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j</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="txt" href="yellow19.html#E6:29"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="yellow19.html#E6:29_2"><i><font color="Green" title="E23">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E8:29_2"/>
<font color="Maroon" title="c2">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c7">G</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E6:29_2"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:29"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="pre_topc.html#K2" title="PRE_TOPC:func.2">Cl</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D13" title="PRE_TOPC:def.13">PRE_TOPC:def 13</a></i>;<br/>


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