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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">a</font> is <a href="lattice6.html#V3" title="LATTICE6:attr.3">completely-meet-irreducible</a> holds <br/>( <font color="Olive" title="b1">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Olive" title="b1">a</font> &amp; ( for <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  ) )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V3" title="LATTICE6:attr.3">completely-meet-irreducible</a> implies ( <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  ) ) )</span><br/>



<b>assume </b><a NAME="E1:32"/>
<font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V3" title="LATTICE6:attr.3">completely-meet-irreducible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  ) )</span><br/>

<a NAME="E2:32"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="lattice6.html#D8" target="_self" title="LATTICE6:def.8">Def8</a></i>;<br/>
<b>set </b><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> ;<br/>
<a NAME="E3:32"/><i><font color="Green" title="E23">A2</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a> 
 
<b>by </b><i><a class="ref" href="lattice6.html#T9" target="_self" title="LATTICE6:th.9">Th9</a></i>;<br/>
<a NAME="E4:32"/>
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  &amp; not <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  &amp; not <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  )</span><br/>

<b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E24">A3</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  )</span><br/>

<b>assume </b><a NAME="E2:32_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a> </span><br/>

<a NAME="E3:32_1"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="lattice6.html#E1:32_1"><i><font color="Green" title="E24">A3</font></i></a></i>;<br/>
<a NAME="E4:32_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E5:32_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E1:32_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:32"/><b>then </b><i><font color="Green" title="E24">A4</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b1">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  &amp; not <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
;<br/>
<a NAME="E6:32"/>
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  )</span><br/>

<b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a> </span><br/>

<a NAME="E2:32_2"/><b>then </b>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">d</font> &amp; <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font> &amp; not <font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> ) )
 
<b>by </b><i><a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/>
<a NAME="E3:32_2"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 
;<br/>
<a NAME="E4:32_2"/><b>then </b>
( <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="lattice6.html#T9" target="_self" title="LATTICE6:th.9">Th9</a>, <a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></i>;<br/>
<b>hence </b><a NAME="E5:32_2"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E2:32"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="lattice6.html#E1:32_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:32"/>
( <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> &amp; ( for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">c</font> <a href="lattice6.html#R1" title="LATTICE6:pred.1">is-upper-neighbour-of</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#K3" title="LATTICE6:func.3">*'</a>  ) )
 <b>by </b><i><a class="txt" href="lattice6.html#E2:32"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="lattice6.html#E3:32"><i><font color="Green" title="E23">A2</font></i></a>, <a class="txt" href="lattice6.html#E5:32"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
