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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">F</font> be  <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">L</font></span>)</span></span>)</span>;<br/>



<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> ;<br/>
<a NAME="E1:26"/><i><font color="Green">E46</font></i>: 
 <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span> ,<font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:26"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span><b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<a NAME="E5:26_1"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>
 
;<br/>
<a NAME="E6:26_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Olive">g</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">g</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">g</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">f</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font><b> such that </b><br/><a NAME="E8:26_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">f</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM1">Element</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">L</font></span>)</span></span>)</span> ;<br/>
<a NAME="E10:26_1"/>
<font color="Maroon">f</font> is   <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T4">WAYBEL_7:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> ;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">i</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> ;<br/>
<a NAME="E13:26_1"/>
<font color="Maroon">i</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E14:26_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E15:26_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:26"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:26"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:26_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">B</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:26_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">B</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<a NAME="E4:26_2"/>
not  <a href="yellow_0.html#K3">Bottom</a> <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">L</font></span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<b>by </b><i><a class="ref" href="waybel_7.html#T8">WAYBEL_7:8</a></i>;<br/>
<a NAME="E5:26_2"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="yellow_1.html#T18">YELLOW_1:18</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:26_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="subset_1.html#T10">SUBSET_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E10:26_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b</font>,<font color="Olive">f</font></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Olive">f</font> is    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">f</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">B</font></span><a href="domain_1.html#K1">]</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i><a class="ref" href="yellow19.html#D4">YELLOW19:def 4</a></i>;<br/>
<a NAME="E12:26_2"/>
<font color="Maroon">i</font> <a href="mcart_1.html#K2">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">B</font>
 
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E13:26_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#K2">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E14:26_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span> 
 ;<br/>

<b>thus </b><a NAME="E15:26_2"/>
verum
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:26"/><b>then </b>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">B</font></span>)</span> where <font color="Olive">B</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_9.html#K2">inf</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> <a href="waybel_9.html#K6">|</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span> : verum </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:26"/>
 <a href="waybel33.html#K1" target="_self">lim_inf</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K1">lim_inf</a> <span class="p1">(<span class="default"><a href="yellow19.html#K3">a_net</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
