<?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="lattice6.html#V2" title="LATTICE6:attr.2">co-noetherian</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<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>;<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="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</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/>
<div><i><font color="Green" title="E25">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a>
 ;<br/><a NAME="E2:35_1"/><b>then </b>
(  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-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#NR2" title="LATTICE6:NR.2">is-lower-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>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> ) )
 <b>by </b><i><a class="ref" href="lattice6.html#T13" target="_self" title="LATTICE6:th.13">Th13</a></i>;<br/><b>hence </b><a NAME="E3:35_1"/>
 ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">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#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">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">b</font> ) )
 ;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><a NAME="E2:35_2"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-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#NR2" title="LATTICE6:NR.2">is-lower-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="c4">b</font> ) )
 ;<br/><a NAME="E3:35_2"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E2:35_2"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/><a NAME="E4:35_2"/><b>then </b><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c4">b</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="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</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="E5:35_2"/>
for <font color="Olive" title="b1">q</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">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">x</font> where <font color="Olive" title="b2">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="Olive" title="b2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) </span>}</span>  holds <br/><font color="Olive" title="b1">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:35_2_1"/>
<font color="Maroon" title="c5">q</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="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</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="E2:35_2_1"/><b>then </b>
 ex <font color="Olive" title="b1">q'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> &amp; <font color="Olive" title="b1">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">q'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> )
 
;<br/>
<a NAME="E3:35_2_1"/><b>then </b>
 ex <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 <br/>( <font color="Maroon" title="c5">q</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="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="lattice6.html#T4" target="_self" title="LATTICE6:th.4">Th4</a></i>;<br/>
<b>hence </b><a NAME="E4:35_2_1"/>
<font color="Maroon" title="c5">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E2:35_2"><i><font color="Green" title="E26">A2</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:35_2"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="lattice3.html#NR8" title="LATTICE3:NR.8">is_greater_than</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="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</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="ref" href="lattice3.html#D17" title="LATTICE3:def.17">LATTICE3:def 17</a></i>;<br/><a NAME="E7:35_2"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:35_2"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="lattice6.html#E4:35_2"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="lattice3.html#T41" title="LATTICE3:th.41">LATTICE3:41</a></i>;<br/><b>hence </b><a NAME="E8:35_2"/>
<font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a>
 <b>by </b><i><a class="ref" href="lattice6.html#D9" target="_self" title="LATTICE6:def.9">Def9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:35"/>
( <font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> iff  ex <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">b</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">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#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">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">b</font> ) ) )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:35"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


</div>
