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

<b>let </b><font color="Maroon">R</font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#V25">/\-complete</a> <a href="waybel32.html#V2" target="_self">order_consistent</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">N</font> be  <a href="waybel_0.html#V10">eventually-directed</a> <a href="waybel_0.html#NM2">net</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/>





<b>thus </b><a NAME="E1:49"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> implies <font color="Maroon">x</font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">N</font> )
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:49"/>
( <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="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">x</font> <a href="waybel_9.html#R3">is_a_cluster_point_of</a> <font color="Maroon">N</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>] <b>means </b>verum;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> =  <a href="yellow_0.html#K2">"/\"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">i</font></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Olive">i</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">a<sub>1</sub></font> </span>}</span> ,<font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span> ;<br/>
<a NAME="E2:49_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 
<b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i>
<b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">j</font>) where <font color="Olive">j</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">j</font>] </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> ;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E5:49_1"/>
for <font color="Olive">G</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>  st <font color="Olive">G</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">G</font> holds <br/><span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>;<br/>

<b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">G</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>assume </b><a NAME="E2:49_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">G</font> = <font color="Maroon">G</font> as    <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="E4:49_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#R2">is_often_in</a> <font color="Maroon">G</font>
 
<b>by </b><i>, <a class="ref" href="waybel_9.html#D9">WAYBEL_9:def 9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/><b>consider </b><font color="Maroon">j1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E2:49_1_1_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">i</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D12">WAYBEL_0:def 12</a></i>;<br/><b>consider </b><font color="Maroon">j2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E4:49_1_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">k</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>  st <font color="Maroon">j2</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Olive">k</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T11">WAYBEL_0:11</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font>;<br/><b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> = <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">E</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span> ;<br/><a NAME="E5:49_1_1_1"/><i><font color="Green">E47</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span>  is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>
 <b>from </b><i><a class="ref" href="domain_1.html#S8">DOMAIN_1:sch 8</a>();<br/></i><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="E7:49_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">j'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="yellow_6.html#D5">YELLOW_6:def 5</a></i>;<br/><a NAME="E8:49_1_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j'</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span> 
 <b>by </b><i><a class="ref" href="waybel32.html#D2" target="_self">Def2</a></i>;<br/><b>then reconsider </b><font color="Maroon">E'</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">k</font>) where <font color="Olive">k</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/><a NAME="E10:49_1_1_1"/><i><font color="Green">E50</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">E'</font>,<font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T76">WAYBEL_0:76</a></i>;<br/><a NAME="E11:49_1_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j1</font> <a href="lattice3.html#R1">is_&lt;=_than</a> <font color="Maroon">E'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8">LATTICE3:def 8</a><br/>

<b>assume </b><a NAME="E1:49_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">E'</font>
 ;<br/>

<b>consider </b><font color="Maroon">k</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E3:49_1_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E4:49_1_1_1_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">k</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j2</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">k'</font> = <font color="Maroon">k</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font> ;<br/>
<b>thus </b><a NAME="E6:49_1_1_1_1"/>
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j1</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, <a class="txt" href="waybel32.html#E1:49_1"><i><font color="Green">E32</font></i></a>, </i>;<br/>


</div><b>end;</b></div><a NAME="E12:49_1_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">N</font> <a href="waybel_0.html#K2">.</a> <font color="Maroon">j1</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">E'</font>,<font color="Maroon">R</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T31">YELLOW_0:31</a></i>;<br/><a NAME="E13:49_1_1_1"/>
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> holds   <a href="waybel_0.html#K5">downarrow</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">a</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i/>;<br/><a NAME="E14:49_1_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">G</font> is <a href="waybel_0.html#V13">upper</a>
 <b>by </b><i>, </i>;<br/><a NAME="E15:49_1_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">E'</font>,<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="waybel32.html#D1" target="_self">Def1</a>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/><a NAME="E16:49_1_1_1"/>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">E'</font>,<font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/><a NAME="E17:49_1_1_1"/><b>then </b>
 <a href="yellow_0.html#K2">"/\"</a> <font color="Maroon">E'</font>,<font color="Maroon">R</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E18:49_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="waybel32.html#E1:49_1"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="waybel_0.html#D20">WAYBEL_0:def 20</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:49_1_1"/><b>then </b>
(  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:49_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:49_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T54">PRE_TOPC:54</a></i>;<br/>
<a NAME="E7:49_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_0.html#K5">downarrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:49_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="waybel11.html#K1">lim_inf</a> <font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T17">WAYBEL_0:17</a></i>;<br/>


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


</div>
