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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V4">complete</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="E1:43"/>
<font color="Maroon">a</font> is <a href="lattice6.html#V6" target="_self">co-atomic</a>
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E3:43"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E3:43"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</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">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">c</font> &amp; <font color="Olive">c</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> &amp; not <font color="Olive">c</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> ) </span>}</span> ;<br/>
<a NAME="E4:43"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> ) </span>}</span>  holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:43_1"/><b>then </b><i><font color="Green">E34</font></i>: 
 ex <font color="Olive">x'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x'</font> &amp; <font color="Olive">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E4:43_1"/>
<font color="Maroon">x</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T45">LATTICES:45</a></i>;<br/>
<a NAME="E5:43_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E4:43"><i><font color="Green">E32</font></i></a>, <a class="txt" href="lattice6.html#E5:43"><i><font color="Green">E35</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:43_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:43"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:43_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E2:43_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:43_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">x</font> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Olive">x</font> &amp; <font color="Olive">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
 <a href="lattices.html#K6">Top</a> <font color="Maroon">L</font> = 
 <a href="lattice3.html#K16">"/\"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">L</font></span>)</span></span><a href="tarski.html#K1">}</a></span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="lattice3.html#T43">LATTICE3:43</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattice6.html#K3" target="_self">*'</a> 
<b>by </b><i>, <a class="txt" href="lattice6.html#E5:43"><i><font color="Green">E35</font></i></a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>
;<br/>
<b>hence </b><a NAME="E7:43"/>
<font color="Maroon">a</font> is <a href="lattice6.html#V3" target="_self">completely-meet-irreducible</a>
 <b>by </b><i>, </i>;<br/>


</div>
