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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>set </b><font color="Maroon" title="c4">X</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E1:34"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12" title="PRE_TOPC:th.12">PRE_TOPC:12</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_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 consider </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><br/><a NAME="E3:34_1"/><i><font color="Green" title="E24">A2</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> )
 <b>by </b><i><a class="ref" href="yellow19.html#T23" target="_self" title="YELLOW19:th.23">Th23</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">S</font> being    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c5">N</font><b> such that </b><br/><a NAME="E5:34_1"/><i><font color="Green" title="E25">A3</font></i>: 
<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> <font color="Maroon" title="c6">S</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:34_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel_9.html#T32" title="WAYBEL_9:th.32">WAYBEL_9:32</a></i>;<br/><b>set </b><font color="Maroon" title="c7">F</font> =  <a href="yellow19.html#K2" title="YELLOW19:func.2">a_filter</a> <font color="Maroon" title="c6">S</font>;<br/><b>consider </b><font color="Maroon" title="c8">G</font> being   <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><b> such that </b><br/><a NAME="E7:34_1"/><i><font color="Green" title="E26">A4</font></i>: 
(  <a href="yellow19.html#K2" title="YELLOW19:func.2">a_filter</a> <font color="Maroon" title="c6">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">G</font> &amp; <font color="Maroon" title="c8">G</font> is <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</a> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T30" title="WAYBEL_7:th.30">WAYBEL_7:30</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">G</font> = <font color="Maroon" title="c8">G</font> as  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</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> <b>by </b><i><a class="txt" href="yellow19.html#E7:34_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c10">G</font> = <font color="Maroon" title="c9">G</font>;<br/><a NAME="E9:34_1"/>
<font color="Maroon" title="c6">S</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="txt" href="yellow19.html#E3:34_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="yellow19.html#T21" target="_self" title="YELLOW19:th.21">Th21</a></i>;<br/><a NAME="E10:34_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow19.html#K2" title="YELLOW19:func.2">a_filter</a> <font color="Maroon" title="c6">S</font>
 ;<br/><b>hence </b><a NAME="E11:34_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">G</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E7:34_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/><a NAME="E12:34_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>  <a href="yellow19.html#K2" title="YELLOW19:func.2">a_filter</a> <font color="Maroon" title="c6">S</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E5:34_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="yellow19.html#T13" target="_self" title="YELLOW19:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E13:34_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="c10">G</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E7:34_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="yellow19.html#T22" target="_self" title="YELLOW19:th.22">Th22</a></i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c5">F</font> being  <a href="waybel_7.html#V3" title="WAYBEL_7:attr.3">ultra</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><b> such that </b><a NAME="E4:34"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">F</font> &amp; <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="c5">F</font>,<font color="Maroon" title="c1">T</font> )
 ;<br/>

<a NAME="E5:34"/>
<font color="Maroon" title="c3">x</font> <a href="waybel_7.html#R1" title="WAYBEL_7:pred.1">is_a_cluster_point_of</a> <font color="Maroon" title="c5">F</font>,<font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="txt" href="yellow19.html#E1:34"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="yellow19.html#E4:34"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="waybel_7.html#T31" title="WAYBEL_7:th.31">WAYBEL_7:31</a></i>;<br/>
<b>hence </b><a NAME="E6:34"/>
<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="txt" href="yellow19.html#E4:34"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="yellow19.html#T27" target="_self" title="YELLOW19:th.27">Th27</a></i>;<br/>


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