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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">N</font> be   <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">T</font>;<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>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>thus </b><a NAME="E1:16"/>
( <font color="Maroon">x</font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">N</font> implies <font color="Maroon">x</font> <a href="waybel_7.html#R1">is_a_cluster_point_of</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>,<font color="Maroon">T</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">O</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font> holds  <font color="Maroon">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Olive">O</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D9">WAYBEL_9:def 9</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D4">WAYBEL_7:def 4</a><br/>

<b>assume </b><a NAME="E2:16_1"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 ;<br/>

<a NAME="E3:16_1"/><b>then </b>
<font color="Maroon">A</font> is    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#T5">CONNSP_2:5</a></i>;<br/>
<a NAME="E4:16_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">A</font>
 
<b>by </b><i/>;<br/>
<b>let </b><font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E5:16_1"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font>
 ;<br/>

<a NAME="E6:16_1"/><b>then </b>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E8:16_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#D11">WAYBEL_0:def 11</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E10:16_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">a</font> = <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E1:16_1_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E12:16_1"/>
not <font color="Maroon">A</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:16"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> holds <br/>for <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">B</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">B</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_7.html#D4">WAYBEL_7:def 4</a><br/>

<b>let </b><font color="Maroon">O</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">x</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D9">WAYBEL_9:def 9</a><br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a><br/>



<b>reconsider </b><font color="Maroon">B</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <span class="p1">(<span class="default"><font color="Maroon">N</font> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span> as    <a href="yellow19.html#M1" target="_self">Subset</a> of <font color="Maroon">T</font>,<font color="Maroon">N</font> <b>by </b><i/>;<br/>
<a NAME="E4:16"/>
<font color="Maroon">N</font> <a href="waybel_0.html#R1">is_eventually_in</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:16"/><b>then </b>
( <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="yellow19.html#K2" target="_self">a_filter</a> <font color="Maroon">N</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> )
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:16"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E10:16"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> )
 <b>by </b><i>, </i>;<br/>
<b>take </b>
<font color="Maroon">j</font>
;<br/>

<a NAME="E11:16"/>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">O</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">O</font>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<b>hence </b><a NAME="E12:16"/>
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">O</font> )
 <b>by </b><i>, </i>;<br/>


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