<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">S</font> be  <a href="compts_1.html#V2">compact</a> <a href="compts_1.html#V3">Hausdorff</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>assume </b><a NAME="E1:97"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds  <font color="Olive">x</font> <a href="waybel_1.html#K4">"/\"</a>  is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

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

<b>assume </b><a NAME="E2:97"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">N</font> is <a href="waybel_0.html#V11">eventually-filtered</a>
 ;<br/>

<b>consider </b><font color="Maroon">c</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E4:97"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c</font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="txt" href="waybel_9.html#E5:97"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E5:97"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#K2" target="_self">inf</a> <font color="Maroon">N</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>thus </b><a NAME="E6:97"/>
 <a href="waybel_9.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">X</font> =  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>;<br/>
<b>reconsider </b><font color="Maroon">d</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> ;<br/>
<a NAME="E2:97_1"/><i><font color="Green">E40</font></i>: 
 <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">d</font>
 
<b>by </b><i>, , , <a class="ref" href="waybel_9.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E3:97_1"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  st  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">d</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Olive">b</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E4:97_1"/>
 <a href="yellow_0.html#R2">ex_inf_of</a>  <a href="relset_1.html#K5">rng</a> the <a href="waybel_0.html#U1">mapping</a> of <font color="Maroon">N</font>,<font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T16">YELLOW_0:16</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D4" target="_self">WAYBEL_9:def 4</a><br/>


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

<a NAME="E7:97"/>
for <font color="Olive">c</font>, <font color="Olive">d</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>  st <font color="Olive">c</font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">N</font> &amp; <font color="Olive">d</font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">N</font> holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>, <font color="Maroon">d</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:97_2"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c</font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 <b>and </b><br/><a NAME="E2:97_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">d</font> <a href="waybel_9.html#R3" target="_self">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 ;<br/>

<b>thus </b><font color="Maroon">c</font> = 
 <a href="waybel_9.html#K2" target="_self">inf</a> <font color="Maroon">N</font>
<b>by </b><i>, , , </i>
<br/>.= 
<font color="Maroon">d</font>
<b>by </b><i>, , <a class="txt" href="waybel_9.html#E7"><i><font color="Green">Lemma55</font></i></a>, </i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:97"/>
 <a href="waybel_9.html#K2" target="_self">inf</a> <font color="Maroon">N</font> <a href="hidden.html#R2">in</a>  <a href="yellow_6.html#K11">Lim</a> <font color="Maroon">N</font>
 <b>by </b><i>, <a class="ref" href="waybel_9.html#T4" target="_self">Th4</a>, </i>;<br/>


</div>
