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<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>set </b><font color="Maroon">R</font> =  <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">Ri</font> = <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K7">~</a> ;<br/>
<div><i><font color="Green">E24</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">L</font> is <a href="lattice6.html#V1" target="_self">noetherian</a>
 ;<br/><a NAME="E2:15_1"/><b>then </b>
 <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="E3:15_1"/><b>then </b><i><font color="Green">E31</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> <a href="lattice3.html#K7">~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i><a class="ref" href="lattice3.html#T8">LATTICE3:8</a></i>;<br/><a NAME="E4:15_1"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K7">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3">LattPOSet</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T20">LATTICE3:20</a></i>;<br/><b>hence </b><a NAME="E5:15_1"/>
<font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattice6.html#V2" target="_self">co-noetherian</a>
 <b>by </b><i>, </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">E32</font></i>: 
<font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattice6.html#V2" target="_self">co-noetherian</a>
 ;<br/><a NAME="E2:15_2"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K7">~</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3">LattPOSet</a> <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T20">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">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> <a href="lattice3.html#K7">~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 <b>by </b><i>, </i>;<br/><a NAME="E4:15_2"/><b>then </b>
 <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><a class="ref" href="lattice3.html#T8">LATTICE3:8</a></i>;<br/><b>hence </b><a NAME="E5:15_2"/>
<font color="Maroon">L</font> is <a href="lattice6.html#V1" target="_self">noetherian</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:15"/>
( <font color="Maroon">L</font> is <a href="lattice6.html#V1" target="_self">noetherian</a> iff <font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a>  is <a href="lattice6.html#V2" target="_self">co-noetherian</a> )
 <b>by </b><i/>;<br/>


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