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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b2">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="Olive" title="b1">A</font> iff  ex <font color="Olive" title="b3">F</font> being <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> st <br/>( <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">F</font> &amp; <font color="Olive" title="b2">x</font> <a href="waybel_7.html#R1" title="WAYBEL_7:pred.1">is_a_cluster_point_of</a> <font color="Olive" title="b3">F</font>,<font color="Maroon" title="c1">T</font> ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">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> iff  ex <font color="Olive" title="b2">F</font> being <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> st <br/>( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">F</font> &amp; <font color="Olive" title="b1">x</font> <a href="waybel_7.html#R1" title="WAYBEL_7:pred.1">is_a_cluster_point_of</a> <font color="Olive" title="b2">F</font>,<font color="Maroon" title="c1">T</font> ) )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> iff  ex <font color="Olive" title="b1">F</font> being <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> st <br/>( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <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="Olive" title="b1">F</font>,<font color="Maroon" title="c1">T</font> ) )</span><br/>





<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">F</font> being <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> st <br/>( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <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="Olive" title="b1">F</font>,<font color="Maroon" title="c1">T</font> ) implies <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> )</span>
<div class="add"><b>assume </b><a NAME="E1:33_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> st <br/>( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> &amp; <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="Olive" title="b1">F</font>,<font color="Maroon" title="c1">T</font> )</span><br/><b>then consider </b><font color="Maroon" title="c4">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="E3:33_1"/><i><font color="Green" title="E22">A1</font></i>: 
( <font color="Maroon" title="c4">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="c4">N</font> )
 <b>by </b><i><a class="ref" href="yellow19.html#T23" target="_self" title="YELLOW19:th.23">Th23</a></i>;<br/><b>set </b><font color="Maroon" title="c5">F</font> =  <a href="yellow19.html#K2" title="YELLOW19:func.2">a_filter</a> <font color="Maroon" title="c4">N</font>;<br/><b>take </b><font color="Maroon" title="c6">F</font> =  <a href="yellow19.html#K2" title="YELLOW19:func.2">a_filter</a> <font color="Maroon" title="c4">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font> &amp; <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="c6">F</font>,<font color="Maroon" title="c1">T</font> )</span><br/><b>thus </b><a NAME="E4:33_1"/>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">F</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:33_1"><i><font color="Green" title="E22">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c6">F</font>,<font color="Maroon" title="c1">T</font></span><br/><b>thus </b><a NAME="E5:33_1"/>
<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="c6">F</font>,<font color="Maroon" title="c1">T</font>
 <b>by </b><i><a class="txt" href="yellow19.html#E3:33_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="yellow19.html#T12" target="_self" title="YELLOW19:th.12">Th12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon" title="c4">F</font> being  <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><b> such that </b><a NAME="E3:33"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> &amp; <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="c4">F</font>,<font color="Maroon" title="c1">T</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">F'</font> = <font color="Maroon" title="c4">F</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/>
<a NAME="E5:33"/>
 <a href="yellow19.html#K2" title="YELLOW19:func.2">a_filter</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c5">F'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font>
 
<b>by </b><i><a class="ref" href="yellow19.html#T15" target="_self" title="YELLOW19:th.15">Th15</a></i>;<br/>
<a NAME="E6:33"/><b>then </b>
(  <a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c5">F'</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>  <a href="yellow19.html#K3" title="YELLOW19:func.3">a_net</a> <font color="Maroon" title="c5">F'</font> )
 
<b>by </b><i><a class="txt" href="yellow19.html#E3:33"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="yellow19.html#T11" target="_self" title="YELLOW19:th.11">Th11</a>, <a class="ref" href="yellow19.html#T12" target="_self" title="YELLOW19:th.12">Th12</a></i>;<br/>
<b>hence </b><a NAME="E7:33"/>
<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="yellow19.html#T23" target="_self" title="YELLOW19:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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