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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/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_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="E3:33_1"/><i><font color="Green">E31</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>set </b><font color="Maroon">F</font> =  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>;<br/><b>take </b><font color="Maroon">F</font> =  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>;<br/><b>thus </b><a NAME="E4:33_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:33_1"/>
<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>, </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">F</font> being  <a href="tex_2.html#V1">proper</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="E2:33"/><i><font color="Green">E36</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#R1">is_a_cluster_point_of</a> <font color="Maroon">F</font>,<font color="Maroon">T</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">F'</font> = <font color="Maroon">F</font> as  <a href="tex_2.html#V1">proper</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> ;<br/>
<a NAME="E4:33"/>
 <a href="yellow19.html#K2" target="_self">a_filter</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F'</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:33"/><b>then </b>
(  <a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F'</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>  <a href="yellow19.html#K3" target="_self">a_net</a> <font color="Maroon">F'</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E6:33"/>
<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/>;<br/>


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