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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">thus </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">z9</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">z9</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">z9</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">z9</font> ) )</span><br/>

<span class="kw">take </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">z9</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">z9</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">z9</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">z9</font> ) )</span><br/>

<a NAME="E1:29_1"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></span>;<br/>
<a NAME="E2:29_1"/><span class="lab"><font color="Green" title="E13">A2</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></span>;<br/>
<a NAME="E3:29_1"/><span class="lab"><font color="Green" title="E14">A3</font></span>: 
<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>
 
;<br/>
<a NAME="E4:29_1"/><span class="lab"><font color="Green" title="E15">A4</font></span>: 
<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>
 
;<br/>
<span class="kw">hence </span><a NAME="E5:29_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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:29_1"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="txt" href="#E2:29_1"><span class="lab"><font color="Green" title="E13">A2</font></span></a>, <a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">z9</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">z9</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">z9</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">z9</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">z9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">z9</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">z9</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">z9</font> )</span><br/>

<a NAME="E6:29_1"/><span class="lab"><font color="Green" title="E16">A5</font></span>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z9</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">z9</font>
 
;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E7:29_1"/><span class="lab"><font color="Green" title="E17">A6</font></span>: 
<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">z9</font>
 <span class="kw">and </span><br/><a NAME="E8:29_1"/><span class="lab"><font color="Green" title="E18">A7</font></span>: 
<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">z9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">z9</font></span><br/>

<a NAME="E9:29_1"/><span class="lab"><font color="Green" title="E19">A8</font></span>: 
 <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">z9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:29_1"><span class="lab"><font color="Green" title="E14">A3</font></span></a>, <a class="txt" href="#E6:29_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E7:29_1"><span class="lab"><font color="Green" title="E17">A6</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>;<br/>
<a NAME="E10:29_1"/>
 <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">z9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29_1"><span class="lab"><font color="Green" title="E15">A4</font></span></a>, <a class="txt" href="#E6:29_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="txt" href="#E8:29_1"><span class="lab"><font color="Green" title="E18">A7</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>;<br/>
<a NAME="E11:29_1"/><span class="kw">then </span>
<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">z9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:29_1"><span class="lab"><font color="Green" title="E19">A8</font></span></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:29_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">z9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:29_1"><span class="lab"><font color="Green" title="E16">A5</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><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>; <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">z9</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">z9</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">z9</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">z9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">z</font> ) )</span><br/>

<span class="kw">take </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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">z9</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">z9</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">z9</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">z9</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"/><span class="lab"><font color="Green" title="E12">A9</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></span>;<br/>
<a NAME="E3:29"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></span>;<br/>
<a NAME="E4:29"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
<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>
 
;<br/>
<a NAME="E5:29"/><span class="lab"><font color="Green" title="E15">A12</font></span>: 
<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>
 
;<br/>
<span class="kw">hence </span><a NAME="E6:29"/>
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:29"><span class="lab"><font color="Green" title="E12">A9</font></span></a>, <a class="txt" href="#E3:29"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">z9</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">z9</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">z9</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">z9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">z9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">z9</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">z9</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">z9</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="E7:29"/><span class="lab"><font color="Green" title="E16">A13</font></span>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z9</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">z9</font>
 
;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E8:29"/><span class="lab"><font color="Green" title="E17">A14</font></span>: 
<font color="Maroon" title="c5">z9</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">and </span><br/><a NAME="E9:29"/><span class="lab"><font color="Green" title="E18">A15</font></span>: 
<font color="Maroon" title="c5">z9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">z9</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="E10:29"/><span class="lab"><font color="Green" title="E19">A16</font></span>: 
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:29"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E8:29"><span class="lab"><font color="Green" title="E17">A14</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>;<br/>
<a NAME="E11:29"/>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:29"><span class="lab"><font color="Green" title="E15">A12</font></span></a>, <a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="txt" href="#E9:29"><span class="lab"><font color="Green" title="E18">A15</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>;<br/>
<a NAME="E12:29"/><span class="kw">then </span>
 <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c5">z9</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:29"><span class="lab"><font color="Green" title="E19">A16</font></span></a>, <a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:29"/>
<font color="Maroon" title="c5">z9</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c4">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:29"><span class="lab"><font color="Green" title="E16">A13</font></span></a>, <a class="ref" href="lattice3.html#T7" target="_self" title="LATTICE3:th.7">Th7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
