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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> &amp;  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> )</span><br/>

<b>thus </b><a NAME="E1:29"/>
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a></span><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" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10" target="_self" title="LATTICE3:def.10">LATTICE3:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> &amp; ( for <font color="Olive" title="b2">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">z'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">z'</font> holds <br/><font color="Olive" title="b1">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">z'</font> ) )</span><br/>

<b>take </b><font color="Maroon" title="c4">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; ( for <font color="Olive" title="b1">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z'</font> holds <br/><font color="Maroon" title="c4">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z'</font> ) )</span><br/>

<a NAME="E1:29_1"/><i><font color="Green" title="E12">A1</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<b>hence </b><a NAME="E2:29_1"/>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">z</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z'</font> holds <br/><font color="Maroon" title="c4">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z'</font></span><br/>

<b>let </b><font color="Maroon" title="c5">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z'</font> implies <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z'</font> )</span><br/>

<a NAME="E3:29_1"/><i><font color="Green" title="E13">A2</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> )
 
;<br/>
<b>assume </b><a NAME="E4:29_1"/>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">z'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z'</font></span><br/>

<a NAME="E5:29_1"/><b>then </b>
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:29_1"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="lattice3.html#E3:29_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>;<br/>
<a NAME="E6:29_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></i>;<br/>
<b>hence </b><a NAME="E7:29_1"/>
<font color="Maroon" title="c4">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">z'</font>
 <b>by </b><i><a class="txt" href="lattice3.html#E3:29_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11" target="_self" title="LATTICE3:def.11">LATTICE3:def 11</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span> st <br/>( <font color="Olive" title="b1">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> &amp; ( for <font color="Olive" title="b2">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b2">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b2">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> holds <br/><font color="Olive" title="b2">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> ) )</span><br/>

<b>take </b><font color="Maroon" title="c4">z</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> &amp; ( for <font color="Olive" title="b1">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> holds <br/><font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> ) )</span><br/>

<a NAME="E2:29"/><i><font color="Green" title="E12">A3</font></i>: 
( <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></i>;<br/>
<b>hence </b><a NAME="E3:29"/>
( <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c4">z</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>  st <font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> holds <br/><font color="Olive" title="b1">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font></span><br/>

<b>let </b><font color="Maroon" title="c5">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font> )</span><br/>

<a NAME="E4:29"/><i><font color="Green" title="E13">A4</font></i>: 
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">z'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> )
 
;<br/>
<b>assume </b><a NAME="E5:29"/>
( <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font></span><br/>

<a NAME="E6:29"/><b>then </b>
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E2:29"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="lattice3.html#E4:29"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>;<br/>
<a NAME="E7:29"/><b>then </b>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></i>;<br/>
<b>hence </b><a NAME="E8:29"/>
<font color="Maroon" title="c5">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font>
 <b>by </b><i><a class="txt" href="lattice3.html#E4:29"><i><font color="Green" title="E13">A4</font></i></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
