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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/>

<b>set </b><font color="Maroon" title="c2">R</font> =  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c3">Ri</font> = <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> ;<br/>
<div><i><font color="Green" title="E9">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattice6.html#V1" title="LATTICE6:attr.1">noetherian</a>
 ;<br/><a NAME="E2:15_1"/><b>then </b>
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 <b>by </b><i><a class="ref" href="lattice6.html#D3" target="_self" title="LATTICE6:def.3">Def3</a></i>;<br/><a NAME="E3:15_1"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
<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 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="lattice3.html#T8" title="LATTICE3:th.8">LATTICE3:8</a></i>;<br/><a NAME="E4:15_1"/>
<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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T20" title="LATTICE3:th.20">LATTICE3:20</a></i>;<br/><b>hence </b><a NAME="E5:15_1"/>
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattice6.html#V2" title="LATTICE6:attr.2">co-noetherian</a>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:15_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="lattice6.html#D4" target="_self" title="LATTICE6:def.4">Def4</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_2"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattice6.html#V2" title="LATTICE6:attr.2">co-noetherian</a>
 ;<br/><a NAME="E2:15_2"/>
<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>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T20" title="LATTICE3:th.20">LATTICE3:20</a></i>;<br/><a NAME="E3:15_2"/><b>then </b>
<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 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="txt" href="lattice6.html#E1:15_2"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="lattice6.html#D4" target="_self" title="LATTICE6:def.4">Def4</a></i>;<br/><a NAME="E4:15_2"/><b>then </b>
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is <a href="wellfnd1.html#V1" title="WELLFND1:attr.1">well_founded</a>
 <b>by </b><i><a class="ref" href="lattice3.html#T8" title="LATTICE3:th.8">LATTICE3:8</a></i>;<br/><b>hence </b><a NAME="E5:15_2"/>
<font color="Maroon" title="c1">L</font> is <a href="lattice6.html#V1" title="LATTICE6:attr.1">noetherian</a>
 <b>by </b><i><a class="ref" href="lattice6.html#D3" target="_self" title="LATTICE6:def.3">Def3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattice6.html#V1" title="LATTICE6:attr.1">noetherian</a> iff <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattice6.html#V2" title="LATTICE6:attr.2">co-noetherian</a> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:15"><i><font color="Green" title="E9">A1</font></i></a></i>;<br/>


</div>
