<?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>let </b><font color="Maroon">a</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">b</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>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font>, <font color="Maroon">y</font> = <font color="Maroon">b</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">c</font> = <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font>;<br/>
<a NAME="E2:12"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22">LATTICES:22</a></i>;<br/>
<a NAME="E3:12"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y</font> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp;  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K4">%</a>  &amp;  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="lattice3.html#K4">%</a>  &amp;  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><font color="Maroon">y</font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> <a href="lattice3.html#K4">%</a>  )
 
;<br/>
<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> as   <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/>
<a NAME="E5:12"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, , <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E6:12"/>
for <font color="Olive">d</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>  st <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font> holds <br/><font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</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="E1:12_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">d</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<a NAME="E3:12_1"/>
( <font color="Maroon">x</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="lattice3.html#K4">%</a>  &amp; <font color="Maroon">y</font> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="lattice3.html#K4">%</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E4:12_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font> )
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<a NAME="E5:12_1"/><b>then </b>
<font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">z</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T6">FILTER_0:6</a></i>;<br/>
<a NAME="E6:12_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">y</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/>
<b>hence </b><a NAME="E7:12_1"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">d</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon">a</font> <a href="lattice3.html#K13">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T22">YELLOW_0:22</a></i>;<br/>


</div>
