<?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>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">d</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Olive">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> ) </span>}</span> ;<br/>
<a NAME="E1:31"/><i><font color="Green">E24</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">d</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Olive">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</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="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">d</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Olive">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</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">X</font> = <span class="p1">{<span class="default"> <font color="Olive">d</font> where <font color="Olive">d</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : ( <font color="Olive">d</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Olive">d</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</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">x</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font>;<br/>
<a NAME="E4:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 
;<br/>
<b>then consider </b><font color="Maroon">x'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:31_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> )
 ;<br/>
<a NAME="E7:31_1"/>
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/><i><font color="Green">E32</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<a NAME="E1:31_2"/>
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/><font color="Olive">q</font> <a href="lattices.html#R3">[=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E2:31_2"/>
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="lattice3.html#D17">LATTICE3:def 17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:31"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR8" target="_self">is_great_than</a>  <a href="xboole_0.html#K1">{}</a>  holds <br/> <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/>
<a NAME="E4:31"/><b>then </b><i><font color="Green">E35</font></i>: 
 <a href="lattice3.html#K15">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E5:31"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_3"><b>proof </b></a><div class="add">

 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> = 
 <a href="yellow_0.html#K1">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow_0.html#D11">YELLOW_0:def 11</a></i>
<br/>.= 
 <a href="lattice3.html#K15">"\/"</a> <a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_0.html#T29">YELLOW_0:29</a></i>
<br/>.= 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font>
<b>by </b><i>, , <a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>
;<br/>
<b>hence </b><a NAME="E2:31_3"/>
<span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</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>
<a NAME="E6:31"/>
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> is <a href="waybel_6.html#V3">join-irreducible</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4"><b>proof </b></a><div class="add">

<a NAME="E1:31_4"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  holds <br/>(  not  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Olive">y</font> or <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> or <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:31_4_1"/>
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:31_4_1"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">x</font> &amp;  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">L'</font> =  <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font> as  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V4">antisymmetric</a> <a href="yellow_0.html#V1">lower-bounded</a>  <a href="orders_2.html#L1">RelStr</a>  ;<br/>
<b>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/>
<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> ;<br/>
<a NAME="E6:31_4_1"/>
( <font color="Maroon">x'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L'</font> or <font color="Maroon">y'</font> <a href="orders_2.html#NR2" target="_self">&gt;=</a>  <a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L'</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44">YELLOW_0:44</a></i>;<br/>
<b>hence </b><a NAME="E7:31_4_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31_4"/>
 <a href="yellow_0.html#K3">Bottom</a> <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> is <a href="waybel_6.html#V3">join-irreducible</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#D3">WAYBEL_6:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:31"/>
(  <a href="lattice6.html#K4" target="_self">*'</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a>  is <a href="waybel_6.html#V3">join-irreducible</a> )
 <b>by </b><i>, <a class="txt" href="lattice6.html#E1:31"><i><font color="Green">E24</font></i></a>, <a class="txt" href="lattice6.html#E1"><i><font color="Green">Lemma25</font></i></a></i>;<br/>


</div>
