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

<b>let </b><font color="Maroon">L</font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:36"/><i><font color="Green">E25</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_0.html#K3">.)</a></span>
 ;<br/>

<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">AA</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span> ;<br/>
<a NAME="E2:36"/><i><font color="Green">E26</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span> 
 ;<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="E3:36_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>thus </b><a NAME="E4:36_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="lopclset.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:36"/>
not <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_2"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">C</font> being   <a href="finsub_1.html#NM1">Finite_Subset</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E2:36_2"/><i><font color="Green">E51</font></i>: 
not <font color="Maroon">C</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i/>;<br/>
<a NAME="E3:36_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">C</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#T32">FINSUB_1:32</a></i>;<br/>
<a NAME="E4:36_2"/><b>then </b>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:36_2"/><b>then </b>
<font color="Maroon">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:36_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T5" target="_self">Th5</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="lattice4.html#K2">FinMeet</a> <font color="Maroon">C</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E9:36_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:36_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 ;<br/>
<a NAME="E12:36_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E13:36_2"/>
not <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</font> ) </span>}</span>  is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">AA</font> = <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">a</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">L</font> <b>by </b><i><a class="ref" href="lopclset.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:36"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">AA</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3"><b>proof </b></a><div class="add">

<a NAME="E1:36_3"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> holds <br/><font color="Olive">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:36_3_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> )
 ;<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="E3:36_3_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:36_3_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E7:36_3_1"/><i><font color="Green">E102</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">d</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E9:36_3_1"/><i><font color="Green">E103</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E10:36_3_1"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lopclset.html#T7" target="_self">Th7</a>, <a class="ref" href="lopclset.html#T9" target="_self">Th9</a>, <a class="ref" href="filter_0.html#T5">FILTER_0:5</a></i>;<br/>
<b>consider </b><font color="Maroon">C</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E12:36_3_1"/><i><font color="Green">E105</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2">FinMeet</a> <font color="Maroon">C</font> &amp; <font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">C</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="lopclset.html#T7" target="_self">Th7</a></i>;<br/>
<b>consider </b><font color="Maroon">E</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E14:36_3_1"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2">FinMeet</a> <font color="Maroon">E</font> &amp; <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">E</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="lopclset.html#T9" target="_self">Th9</a></i>;<br/>
<a NAME="E15:36_3_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2">FinMeet</a> <span class="p1">(<span class="default"><font color="Maroon">C</font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">E</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="lattice4.html#T30">LATTICE4:30</a></i>;<br/>
<a NAME="E16:36_3_1"/>
( <font color="Maroon">C</font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">E</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Maroon">C</font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">E</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="xboole_1.html#T8">XBOOLE_1:8</a>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/>
<a NAME="E17:36_3_1"/><b>then </b>
<font color="Maroon">c</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E18:36_3_1"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 <b>by </b><i><a class="ref" href="lopclset.html#T6" target="_self">Th6</a>, <a class="ref" href="lopclset.html#T8" target="_self">Th8</a>, <a class="ref" href="lopclset.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:36_3"/>
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> &amp; <font color="Olive">p</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font> holds <br/><font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_3_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<b>assume </b><a NAME="E1:36_3_2"/><i><font color="Green">E110</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font> )
 ;<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="E3:36_3_2"/><i><font color="Green">E112</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E5:36_3_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E6:36_3_2"/>
<font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 
<b>by </b><i>, <a class="ref" href="lopclset.html#T6" target="_self">Th6</a>, <a class="ref" href="lopclset.html#T7" target="_self">Th7</a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E8:36_3_2"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> ) )
 <b>by </b><i><a class="ref" href="lopclset.html#T7" target="_self">Th7</a></i>;<br/>
<b>thus </b><a NAME="E9:36_3_2"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 <b>by </b><i><a class="ref" href="lopclset.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:36_3"/>
<font color="Maroon">AA</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">L</font>
 <b>by </b><i>, <a class="ref" href="filter_0.html#T9">FILTER_0:9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:36"/>
<font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">AA</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:36_4"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:36_4"/><b>then </b><i><font color="Green">E116</font></i>: 
( <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <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="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:36_4"/>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">C</font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span> as    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/>
<a NAME="E5:36_4"/><i><font color="Green">E117</font></i>: 
( the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V1">commutative</a> &amp; the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> is <a href="binop_1.html#V2">associative</a> )
 
<b>by </b><i><a class="ref" href="lattice2.html#T31">LATTICE2:31</a>, <a class="ref" href="lattice2.html#T32">LATTICE2:32</a></i>;<br/>
<i><font color="Green">E118</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T35">FUNCT_1:35</a></i>
<br/>.= 
the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="setwiseo.html#K7">$$</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>
<b>by </b><i>, <a class="ref" href="lopclset.html#T6" target="_self">Th6</a>, <a class="ref" href="setwiseo.html#T26">SETWISEO:26</a></i>
<br/>.= 
 <a href="lattice2.html#K3">FinMeet</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice2.html#D4">LATTICE2:def 4</a></i>
<br/>.= 
 <a href="lattice2.html#K3">FinMeet</a> <font color="Maroon">C</font>,<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="grcat_1.html#D11">GRCAT_1:def 11</a></i>
<br/>.= 
 <a href="lattice4.html#K2">FinMeet</a> <font color="Maroon">C</font>
<b>by </b><i><a class="ref" href="lattice4.html#D13">LATTICE4:def 13</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">a</font> =  <a href="lattice4.html#K2">FinMeet</a> <font color="Maroon">C</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E8:36_4"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:36_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 <b>by </b><i><a class="ref" href="lopclset.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:36"/><b>then </b>
<span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_0.html#K3">.)</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">AA</font>
 
<b>by </b><i><a class="ref" href="lopclset.html#T5" target="_self">Th5</a>, <a class="ref" href="filter_0.html#D5">FILTER_0:def 5</a></i>;<br/>
<a NAME="E8:36"/><b>then </b>
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <font color="Maroon">AA</font>
 
<b>by </b><i/>;<br/>
<a NAME="E9:36"/><b>then </b>
 ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">d</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp;  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">d</font> ) )
 
;<br/>
<b>then consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E11:36"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span>  &amp; <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 ;<br/>
<a NAME="E12:36"/>
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/>
<a NAME="E13:36"/><b>then </b>
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font></span>)</span> where <font color="Olive">C</font> is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> : ( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) </span>}</span> 
 
<b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>
<a NAME="E14:36"/><b>then </b>
 ex <font color="Olive">C</font> being  <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>(  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="lattice4.html#K2">FinMeet</a> <font color="Olive">C</font> &amp; <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp; <font color="Olive">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 
;<br/>
<b>hence </b><a NAME="E15:36"/>
 ex <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finsub_1.html#NM1">Finite_Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font> &amp;  <a href="lattice4.html#K2">FinMeet</a> <font color="Olive">B</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 ;<br/>


</div>
