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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="subset_1.html#V1" title="SUBSET_1: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/> 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="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="Olive" title="b1">F</font> iff <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="b1">F</font>,<font color="Maroon" title="c1">T</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   <a href="subset_1.html#V1" title="SUBSET_1: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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="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="c2">F</font> iff <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="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">T</font> )</span><br/>



<a NAME="E1:25"/>
<font color="Maroon" title="c2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <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="c2">F</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow19.html#T15" target="_self" title="YELLOW19:th.15">Th15</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:25"/>
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="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="c2">F</font> iff <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="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow19.html#T12" target="_self" title="YELLOW19:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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