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<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="E1:96"/><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="E2:96"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">N</font> is <a href="waybel_0.html#V10">eventually-directed</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="E4:96"/><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#E5:96"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E5:96"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1">sup</a> <font color="Maroon">N</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, , , <a class="ref" href="waybel_9.html#T2" target="_self">Th2</a></i>;<br/>
<b>thus </b><a NAME="E6:96"/>
 <a href="waybel_9.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">N</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_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="E2:96_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#R2">is_&lt;=_than</a> <font color="Maroon">d</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, , , </i>;<br/>
<a NAME="E3:96_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#R2">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">d</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font>
 
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, , </i>;<br/>
<b>hence </b><a NAME="E4:96_1"/>
 <a href="yellow_0.html#R1">ex_sup_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="waybel_9.html#T4" target="_self">Th4</a>, <a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_9.html#D3" target="_self">WAYBEL_9:def 3</a><br/>


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

<a NAME="E7:96"/>
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="96_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="E1:96_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="E2:96_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_2.html#K1">sup</a> <font color="Maroon">N</font>
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, , <a class="ref" href="waybel_9.html#T4" target="_self">Th4</a>, <a class="ref" href="waybel_9.html#T2" target="_self">Th2</a></i>
<br/>.= 
<font color="Maroon">d</font>
<b>by </b><i><a class="txt" href="waybel_9.html#E5"><i><font color="Green">Lemma49</font></i></a>, , , <a class="ref" href="waybel_9.html#T2" target="_self">Th2</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:96"/>
 <a href="waybel_2.html#K1">sup</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>, , </i>;<br/>


</div>
