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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <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> 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="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> holds <br/>( <font color="Olive" title="b1">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> iff <font color="Olive" title="b1">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  is <a href="waybel_6.html#V3" title="WAYBEL_6:attr.3">join-irreducible</a> )</span><br/><b>let </b><font color="Maroon" title="c2">a</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="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> implies ( <font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> iff <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  is <a href="waybel_6.html#V3" title="WAYBEL_6:attr.3">join-irreducible</a> ) )</span><br/>



<b>assume </b><a NAME="E1:40"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <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="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> iff <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  is <a href="waybel_6.html#V3" title="WAYBEL_6:attr.3">join-irreducible</a> )</span><br/>

<b>then consider </b><font color="Maroon" title="c3">b</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="E3:40"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c3">b</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="ref" href="lattice6.html#T8" target="_self" title="LATTICE6:th.8">Th8</a></i>;<br/>
<a NAME="E4:40"/><i><font color="Green" title="E31">A2</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c3">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; ( for <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 <font color="Maroon" title="c3">b</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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; not <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> ) )
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></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" title="E32">A3</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  is <a href="waybel_6.html#V3" title="WAYBEL_6:attr.3">join-irreducible</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a></span><br/><a NAME="E2:40_1"/>
for <font color="Olive" title="b1">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="b1">d</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">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" title="c4">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="c4">d</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )</span><br/>

<b>assume </b><a NAME="E1:40_1_1"/><i><font color="Green" title="E33">A4</font></i>: 
<font color="Maroon" title="c4">d</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"> <font color="Maroon" title="c4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font></span><br/>

<a NAME="E2:40_1_1"/><b>then </b><i><font color="Green" title="E34">A5</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; ( for <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 <font color="Maroon" title="c4">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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; not <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> holds <br/><font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> ) )
 
<b>by </b><i><a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></i>;<br/>
<b>assume </b><a NAME="E3:40_1_1"/>
<font color="Maroon" title="c4">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E4:40_1_1"/><b>then </b><i><font color="Green" title="E35">A6</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="lattice6.html#E1:40_1_1"><i><font color="Green" title="E33">A4</font></i></a>, <a class="ref" href="lattice6.html#T2" target="_self" title="LATTICE6:th.2">Th2</a></i>;<br/>
<a NAME="E5:40_1_1"/><i><font color="Green" title="E36">A7</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c4">d</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> &amp; <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></i>;<br/>
<a NAME="E6:40_1_1"/><b>then </b>
<font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">d</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="lattice6.html#E4:40_1_1"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="lattice6.html#E33"><i><font color="Green" title="E26">Lm8</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" title="E31">A2</font></i></a>, <a class="txt" href="lattice6.html#E1:40_1"><i><font color="Green" title="E32">A3</font></i></a>, <a class="txt" href="lattice6.html#E2:40_1_1"><i><font color="Green" title="E34">A5</font></i></a>, <a class="txt" href="lattice6.html#E5:40_1_1"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="waybel_6.html#D3" title="WAYBEL_6:def.3">WAYBEL_6:def 3</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><b>hence </b><a NAME="E3:40_1"/>
<font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:40"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="lattice6.html#T15" target="_self" title="LATTICE6:th.15">Th15</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>
<b>hence </b><a NAME="E6:40"/>
( <font color="Maroon" title="c2">a</font> is <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> iff <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  is <a href="waybel_6.html#V3" title="WAYBEL_6:attr.3">join-irreducible</a> )
 <b>by </b><i><a class="ref" href="lattice6.html#T18" target="_self" title="LATTICE6:th.18">Th18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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