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

<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>thus </b><a NAME="E1:29"/>
 <a href="lattice3.html#K3" target="_self">LattPOSet</a> <font color="Maroon">L</font> is <a href="lattice3.html#V1" target="_self">with_suprema</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_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" target="_self">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" target="_self">LattPOSet</a> <font color="Maroon">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10" target="_self">LATTICE3:def 10</a><br/>

<b>take </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="lattice3.html#K4" target="_self">%</a> ;<br/>

<a NAME="E1:29_1"/><i><font color="Green">E18</font></i>: 
(  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> &amp;  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> &amp;  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> &amp;  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<b>hence </b><a NAME="E2:29_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> )
 <b>by </b><i/>;<br/>

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

<a NAME="E3:29_1"/><i><font color="Green">E22</font></i>: 
(  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font></span>)</span> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> )
 
;<br/>
<b>assume </b><a NAME="E4:29_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font> &amp; <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z'</font> )
 ;<br/>

<a NAME="E5:29_1"/><b>then </b>
(  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> &amp;  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> )
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E6:29_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<b>hence </b><a NAME="E7:29_1"/>
<font color="Maroon">z</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z'</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>

<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" target="_self">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" target="_self">LattPOSet</a> <font color="Maroon">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11" target="_self">LATTICE3:def 11</a><br/>

<b>take </b><font color="Maroon">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span></span>)</span> <a href="lattice3.html#K4" target="_self">%</a> ;<br/>

<a NAME="E2:29"/><i><font color="Green">E25</font></i>: 
( <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> &amp;  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> &amp;  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23">LATTICES:23</a></i>;<br/>
<b>hence </b><a NAME="E3:29"/>
( <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>

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

<a NAME="E4:29"/><i><font color="Green">E26</font></i>: 
(  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font></span>)</span> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> )
 
;<br/>
<b>assume </b><a NAME="E5:29"/>
( <font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">z'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 ;<br/>

<a NAME="E6:29"/><b>then </b>
(  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font> &amp;  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="lattice3.html#D1" target="_self">Def1</a>, </i>;<br/>
<a NAME="E7:29"/><b>then </b>
 <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">z'</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">x</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="filter_0.html#T7">FILTER_0:7</a></i>;<br/>
<b>hence </b><a NAME="E8:29"/>
<font color="Maroon">z'</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self">Def1</a>, </i>;<br/>


</div>
