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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/>



<a NAME="E1:35"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T52">PRE_TOPC:52</a></i>;<br/>
<b>hence </b><a NAME="E2:35"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> implies for <font color="Olive">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>  st <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="waybel_7.html#R1">is_a_cluster_point_of</a> <font color="Olive">F</font>,<font color="Maroon">T</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E3:35"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">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>  st <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> holds <br/>for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">x</font> <a href="waybel_7.html#R1">is_a_cluster_point_of</a> <font color="Olive">F</font>,<font color="Maroon">T</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<a NAME="E4:35"/>
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:35_1"/><i><font color="Green">E36</font></i>: 
<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 reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:35_1"/>
 ex <font color="Olive">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> st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> &amp; <font color="Maroon">y</font> <a href="waybel_7.html#R1">is_a_cluster_point_of</a> <font color="Olive">F</font>,<font color="Maroon">T</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:35_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:35"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>


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