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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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</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>set </b><font color="Maroon">X</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E1:34"/><i><font color="Green">E31</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 ;<br/><b>then consider </b><font color="Maroon">N</font> being   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:34_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">N</font> )
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">S</font> being    <a href="yellow_6.html#M2">subnet</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E5:34_1"/><i><font color="Green">E37</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">S</font>
 <b>by </b><i>, <a class="ref" href="waybel_9.html#T32">WAYBEL_9:32</a></i>;<br/><b>set </b><font color="Maroon">F</font> =  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">S</font>;<br/><b>consider </b><font color="Maroon">G</font> being   <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E7:34_1"/><i><font color="Green">E40</font></i>: 
(  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">S</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">G</font> is <a href="waybel_7.html#V3">ultra</a> )
 <b>by </b><i><a class="ref" href="waybel_7.html#T30">WAYBEL_7:30</a></i>;<br/><b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as  <a href="waybel_7.html#V3">ultra</a> <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span> <b>by </b><i><a class="ref" href="yellow19.html#T2" target="_self">Th2</a></i>;<br/><b>take </b><font color="Maroon">G</font> = <font color="Maroon">G</font>;<br/><a NAME="E9:34_1"/>
<font color="Maroon">S</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/><a NAME="E10:34_1"/><b>then </b>
<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">S</font>
 ;<br/><b>hence </b><a NAME="E11:34_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="yellow19.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E12:34_1"/>
<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">S</font>,<font color="Maroon">T</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E13:34_1"/>
<font color="Maroon">x</font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a> <font color="Maroon">G</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow19.html#T2" target="_self">Th2</a>, </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">F</font> being  <a href="waybel_7.html#V3">ultra</a> <a href="waybel_0.html#NM4">Filter</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3">BoolePoset</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">T</font></span>)</span></span>)</span><b> such that </b><a NAME="E3:34"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">x</font> <a href="waybel_7.html#R2">is_a_convergence_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font> )
 ;<br/>

<a NAME="E4:34"/>
<font color="Maroon">x</font> <a href="waybel_7.html#R1">is_a_cluster_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font>
 
<b>by </b><i>, , <a class="ref" href="waybel_7.html#T31">WAYBEL_7:31</a></i>;<br/>
<b>hence </b><a NAME="E5:34"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i>, </i>;<br/>


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