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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice3.html#V4">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span> ;<br/>
<a NAME="E1:30"/><i><font color="Green">E21</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:30_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> : (  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> ) </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E4:30_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:30_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E7:30_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T45">LATTICES:45</a></i>;<br/>
<b>hence </b><a NAME="E8:30_1"/>
contradiction
 <b>by </b><i><a class="txt" href="lattice6.html#E6:30_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/><i><font color="Green">E22</font></i>: 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<a NAME="E1:30_2"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E2:30_2"/>
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="lattice3.html#D16">LATTICE3:def 16</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:30"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="lattice3.html#R3">is_less_than</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="lattices.html#T45">LATTICES:45</a></i>;<br/>
<a NAME="E4:30"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="lattice3.html#K16">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="lattice6.html#E2:30"><i><font color="Green">E22</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E5:30"/>
<span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><b>proof </b></a><div class="add">

 <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="yellow_0.html#K2">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D12">YELLOW_0:def 12</a></i>
<br/>.= 
 <a href="lattice3.html#K16">"/\"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="yellow_0.html#T29">YELLOW_0:29</a></i>
<br/>.= 
 <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="lattice6.html#E2:30"><i><font color="Green">E22</font></i></a>, <a class="txt" href="lattice6.html#E3:30"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>
;<br/>
<b>hence </b><a NAME="E2:30_3"/>
<span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K4">Top</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:30"/>
( <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice6.html#K3" target="_self">*'</a>  <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K4">%</a>  is <a href="waybel_6.html#V2">meet-irreducible</a> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:30"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lattice6.html#E4:30"><i><font color="Green">E24</font></i></a>, <a class="ref" href="waybel_6.html#T10">WAYBEL_6:10</a></i>;<br/>


</div>
