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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice6.html#V2" title="LATTICE6:attr.2">co-noetherian</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>, <font color="Olive" title="b2">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="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">d</font> holds <br/> ex <font color="Olive" title="b3">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="Olive" title="b2">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b3">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Olive" title="b1">a</font> )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">d</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="c3">d</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="c3">d</font> implies  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="c3">d</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> ) )</span><br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c3">d</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="c3">d</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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="c3">d</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> )</span><br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>] <b>means </b>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> $1</span>)</span> <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>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> $1</span>)</span> implies  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/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> $1</span>)</span> <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> ) );<br/>
<a NAME="E2:22"/><i><font color="Green" title="E15">A2</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 
<b>by </b><i><a class="ref" href="lattice6.html#D4" target="_self" title="LATTICE6:def.4">Def4</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green" title="E16">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>  st ( for <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">y</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>; <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">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>] )</span><br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green" title="E17">A4</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">x</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">y</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>]</span><br/>

<a NAME="E2:22_1"/>
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> implies  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/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green" title="E18">A5</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  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/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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> )</span><br/>

<i><font color="Green" title="E19">A6</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  = 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
<br/>.= 
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
;<br/>
<i><font color="Green" title="E20">A7</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  = 
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_3"/>
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> or  not  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_3_1"><b>suppose </b></a><a NAME="E1:22_1_1_3_1"/>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-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">  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/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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> )</span><br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_3_1"/>
 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/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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> )
 ; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_3_2"><b>suppose </b></a><a NAME="E1:22_1_1_3_2"/>
not  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-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">  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/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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> )</span><br/><div class="add"><b>then consider </b><font color="Maroon" title="c5">c</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><br/><a NAME="E3:22_1_1_3_2"/><i><font color="Green" title="E21">A8</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c5">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; not <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; not <font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:22_1_1"><i><font color="Green" title="E18">A5</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/><i><font color="Green" title="E22">A9</font></i>:  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span>)</span> = 
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c5">c</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
;<br/><a NAME="E5:22_1_1_3_2"/>
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E2:22_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="lattice6.html#E3:22_1_1_3_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/><a NAME="E6:22_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:22_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="ref" href="lattice3.html#T9" title="LATTICE3:th.9">LATTICE3:9</a></i>;<br/><a NAME="E7:22_1_1_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">c'</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><br/><a NAME="E9:22_1_1_3_2"/><i><font color="Green" title="E23">A10</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">c</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">c'</font> &amp; <font color="Maroon" title="c6">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="txt" href="lattice6.html#E1:22_1"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="lattice6.html#E3:22_1_1_3_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="lattice6.html#E4:22_1_1_3_2"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/><b>thus </b><a NAME="E10:22_1_1_3_2"/>
 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/>(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">x</font></span>)</span> <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="txt" href="lattice6.html#E3:22_1_1_3_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="lattice6.html#E4:22_1_1_3_2"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="lattice6.html#E9:22_1_1_3_2"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</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></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">x</font>]
 ; <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="E4:22"/><i><font color="Green" title="E17">A11</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>]
 
<b>from </b><i><a class="ref" href="wellfnd1.html#S3" title="WELLFND1:sch.3">WELLFND1:sch 3</a>(<a class="txt" href="lattice6.html#E3:22"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="lattice6.html#E2:22"><i><font color="Green" title="E15">A2</font></i></a>);<br/></i>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span></span>)</span> = 
 <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D4" title="LATTICE3:def.4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">d</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D7" title="LATTICE3:def.7">LATTICE3:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">d</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D6" title="LATTICE3:def.6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c3">d</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:22"/>
 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="c3">d</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="txt" href="lattice6.html#E1:22"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="lattice6.html#E4:22"><i><font color="Green" title="E17">A11</font></i></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>
