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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice6.html#V1" target="_self">noetherian</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:21"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>] <b>means </b>( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a<sub>1</sub></font> implies  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="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">c</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> ) );<br/>
<a NAME="E2:21"/><i><font color="Green">E31</font></i>: 
 <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font> is <a href="wellfnd1.html#V1">well_founded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:21"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st ( for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Olive">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:21_1"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">y</font>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>]
 ;<br/>

<a NAME="E2:21_1"/>
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> implies  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="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp; <font color="Olive">c</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:21_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> )
 ;<br/>

<i><font color="Green">E36</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
<font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:21_1_1_2"/>
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> or  not  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2_1"><b>suppose </b></a><a NAME="E1:21_1_1_2_1"/>
 <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:21_1_1_2_1"/>
 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="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp; <font color="Olive">c</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="21_1_1_2_2"><b>suppose </b></a><a NAME="E1:21_1_1_2_2"/>
not  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font>
 ;<br/><div class="add"><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="E3:21_1_1_2_2"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp; not <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/><i><font color="Green">E38</font></i>:  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattice3.html#K4">%</a> </span>)</span> = 
<font color="Maroon">c</font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon">c</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/><a NAME="E5:21_1_1_2_2"/>
<font color="Maroon">c</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/><a NAME="E6:21_1_1_2_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattice3.html#K4">%</a> </span>)</span>,<font color="Maroon">x</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></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="E8:21_1_1_2_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">c'</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E9:21_1_1_2_2"/>
<font color="Maroon">c'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/><b>hence </b><a NAME="E10:21_1_1_2_2"/>
 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="lattices.html#R3">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">x</font> &amp; <font color="Olive">c</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:21"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>]
 
<b>from </b><i><a class="ref" href="wellfnd1.html#S3">WELLFND1:sch 3</a>(, );<br/></i>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice3.html#K4">%</a> </span>)</span> = 
<font color="Maroon">d</font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<font color="Maroon">d</font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:21"/>
 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="lattices.html#R3">[=</a> <font color="Maroon">d</font> &amp; <font color="Olive">c</font> <a href="lattice6.html#R1" target="_self">is-upper-neighbour-of</a> <font color="Maroon">a</font> )
 <b>by </b><i>, </i>;<br/>


</div>
