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

<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="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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> is  <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a>  holds <br/>(  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Olive" title="b1">a</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Olive" title="b1">a</font> &amp; ( for <font color="Olive" title="b2">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="Olive" title="b2">c</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Olive" title="b2">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Olive" title="b1">a</font> ) )</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="Red">::  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>  implies (  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</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="Olive" title="b1">c</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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> ) ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">X</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> ;<br/>
<a NAME="E1:33"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
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  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</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>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">c</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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; not <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> implies <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:33_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
 <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E2:33_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:33_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E4:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:33_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/>
<a NAME="E5:33_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:33_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:33_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:33"/>
<font color="Maroon" title="c2">a</font> is  <a href="lattice6.html#V4" title="LATTICE6:attr.4">completely-join-irreducible</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</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="Olive" title="b1">c</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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> ) )</span><br/>

<a NAME="E3:33"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A4</font></span>: 
 <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice6.html#D9" target="_self" title="LATTICE6:def.9">Def9</a></span>;<br/>
<a NAME="E4:33"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
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="Olive" title="b1">c</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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">c</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">c</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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:33_2"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c4">c</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font></span><br/>

<a NAME="E2:33_2"/><span class="kw">then </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">c</font> &amp; <font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</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/>
<a NAME="E3:33_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">x</font> where <font color="Olive" title="b1">x</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> : ( <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font> &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">a</font> ) </span>}</span> 
 
;<br/>
<a NAME="E4:33_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c4">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#T38" title="LATTICE3:th.38">LATTICE3:38</a></span>;<br/>
<a NAME="E5:33_2"/>
 <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice6.html#T9" target="_self" title="LATTICE6:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:33_2"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E1:33_2"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E4:33_2"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:33"/>
 <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice6.html#T9" target="_self" title="LATTICE6:th.9">Th9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:33"/>
(  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> <a href="lattice6.html#NR2" title="LATTICE6:NR.2">is-lower-neighbour-of</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="Olive" title="b1">c</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">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice6.html#K4" title="LATTICE6:func.4">*'</a> <font color="Maroon" title="c2">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E1:33"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E4:33"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="lattice6.html#D5" target="_self" title="LATTICE6:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
