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

<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>

<b>thus </b><a NAME="E1:28"/>
( <font color="Maroon" title="c1">A</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a> implies <font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a>  is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:28_1"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  ex <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">c</font> &amp; ( for <font color="Olive" title="b4">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st <font color="Olive" title="b1">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">c'</font> &amp; <font color="Olive" title="b2">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">c'</font> holds <br/><font color="Olive" title="b3">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b4">c'</font> ) )
 ; <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><br/>

<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"><font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </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><br/>

<b>consider </b><font color="Maroon" title="c4">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E13">A2</font></i>: 
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <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">c</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <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">c</font> &amp; ( for <font color="Olive" title="b1">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>  st  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <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">c'</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <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">c'</font> holds <br/><font color="Maroon" title="c4">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">c'</font> ) )
 <b>by </b><i><a class="txt" href="lattice3.html#E1:28_1"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">c</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> ;<br/>

<a NAME="E4:28_1"/><i><font color="Green" title="E14">A3</font></i>: 
( <font color="Maroon" title="c4">c</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</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#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</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#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font> )
 
;<br/>
<b>hence </b><a NAME="E5:28_1"/>
( <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> )
 <b>by </b><i><a class="txt" href="lattice3.html#E3:28_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="lattice3.html#T9" target="_self" title="LATTICE3:th.9">Th9</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>;<br/>

<a NAME="E6:28_1"/><i><font color="Green" title="E15">A4</font></i>: 
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">z'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">z'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">z'</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">z'</font> )
 
;<br/>
<b>assume </b><a NAME="E7:28_1"/>
( <font color="Maroon" title="c6">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="c6">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">y</font> )
 ;<br/>

<a NAME="E8:28_1"/><b>then </b>
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">z'</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">z'</font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E4:28_1"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="lattice3.html#E6:28_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="lattice3.html#T9" target="_self" title="LATTICE3:th.9">Th9</a></i>;<br/>
<a NAME="E9:28_1"/><b>then </b>
<font color="Maroon" title="c4">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c6">z'</font>
 
<b>by </b><i><a class="txt" href="lattice3.html#E3:28_1"><i><font color="Green" title="E13">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:28_1"/>
<font color="Maroon" title="c6">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#E6:28_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="lattice3.html#T9" target="_self" title="LATTICE3:th.9">Th9</a></i>;<br/>


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

<b>assume </b><a NAME="E2:28"/><i><font color="Green" title="E12">A5</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>  ex <font color="Olive" title="b3">z</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span> st <br/>( <font color="Olive" title="b3">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b3">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> &amp; ( for <font color="Olive" title="b4">z'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span>  st <font color="Olive" title="b4">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">x</font> &amp; <font color="Olive" title="b4">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b4">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">z</font> ) )
 ; <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><br/>

<b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>; <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><br/><b>let </b><font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>



<b>consider </b><font color="Maroon" title="c4">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </span>)</span><b> such that </b><br/><a NAME="E4:28"/><i><font color="Green" title="E13">A6</font></i>: 
( <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">a</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  &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">b</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  &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"><font color="Maroon" title="c1">A</font> <a href="lattice3.html#K7" title="LATTICE3:func.7">~</a> </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">a</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  &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">b</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  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> ) )
 <b>by </b><i><a class="txt" href="lattice3.html#E2:28"><i><font color="Green" title="E12">A5</font></i></a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">c</font> =  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">z</font>;<br/>

<a NAME="E5:28"/><i><font color="Green" title="E14">A7</font></i>: 
(  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">z</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">z</font></span>)</span> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> &amp;  <a href="lattice3.html#K9" title="LATTICE3:func.9">~</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  )
 
;<br/>
<b>hence </b><a NAME="E6:28"/>
( <font color="Maroon" title="c2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="lattice3.html#E4:28"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="lattice3.html#T9" target="_self" title="LATTICE3:th.9">Th9</a></i>;<br/>

<b>let </b><font color="Maroon" title="c6">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>

<b>assume </b><a NAME="E7:28"/>
( <font color="Maroon" title="c2">a</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c'</font> &amp; <font color="Maroon" title="c3">b</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c'</font> )
 ;<br/>

<a NAME="E8:28"/><b>then </b>
( <font color="Maroon" title="c6">c'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">a</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  &amp; <font color="Maroon" title="c6">c'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c3">b</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  )
 
<b>by </b><i><a class="ref" href="lattice3.html#T9" target="_self" title="LATTICE3:th.9">Th9</a></i>;<br/>
<a NAME="E9:28"/><b>then </b>
<font color="Maroon" title="c6">c'</font> <a href="lattice3.html#K8" title="LATTICE3:func.8">~</a>  <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#E4:28"><i><font color="Green" title="E13">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:28"/>
<font color="Maroon" title="c5">c</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">c'</font>
 <b>by </b><i><a class="txt" href="lattice3.html#E5:28"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="lattice3.html#T9" target="_self" title="LATTICE3:th.9">Th9</a></i>;<br/>


</div>
