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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">assume </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:40"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice6.html#T8" target="_self" title="LATTICE6:th.8">Th8</a></span>;<br/>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:40_1"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E1:40_1_1"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>;<br/>
<a NAME="E2:40_1_1"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D3" title="LATTICE3:def.3">LATTICE3:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:40_1_1"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E4:40_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:40_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:40_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:40_1_1"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="lattice6.html#T2" target="_self" title="LATTICE6:th.2">Th2</a></span>;<br/>
<a NAME="E7:40_1_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="lattice6.html#E33"><span class="lab"><font color="Green" title="E26">Lm8</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:40_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E1:40_1"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:40_1_1"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E1:40_1_1"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E2:40_1_1"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="waybel_6.html#D3" title="WAYBEL_6:def.3">WAYBEL_6:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="ref" href="lattice6.html#T15" target="_self" title="LATTICE6:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice6.html#T18" target="_self" title="LATTICE6:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
