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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</a> <a href="lattice6.html#V2" target="_self">co-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>assume </b><a NAME="E1:40"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:40"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">b</font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:40"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> ) )
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:40_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  is <a href="waybel_6.html#V3">join-irreducible</a>
 ;<br/><a NAME="E2:40_1"/>
for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">d</font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">a</font> holds <br/><font color="Olive">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1_1"><b>proof </b></a><div class="add">

<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="E1:40_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">d</font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:40_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> ) )
 
<b>by </b><i/>;<br/>
<b>assume </b><a NAME="E3:40_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E4:40_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:40_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">d</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>
<a NAME="E6:40_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">d</font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattice3.html#K4">%</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:40_1_1"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E7:40_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice6.html#E4:40"><i><font color="Green">E31</font></i></a>, <a class="txt" href="lattice6.html#E1:40_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="lattice6.html#E1:40_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="lattice6.html#E4:40_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel_6.html#D3">WAYBEL_6:def 3</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:40_1"/>
<font color="Maroon">a</font> is <a href="lattice6.html#V4" target="_self">completely-join-irreducible</a>
 <b>by </b><i>, <a class="txt" href="lattice6.html#E3:40"><i><font color="Green">E24</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:40"/>
( <font color="Maroon">a</font> is <a href="lattice6.html#V4" target="_self">completely-join-irreducible</a> iff <font color="Maroon">a</font> <a href="lattice3.html#K4">%</a>  is <a href="waybel_6.html#V3">join-irreducible</a> )
 <b>by </b><i/>;<br/>


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