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

<a NAME="E1:65_1_1"/>
( <font color="Maroon">A</font> is <a href="lattice3.html#V1" target="_self">with_suprema</a> &amp; <font color="Maroon">A</font> is <a href="lattice3.html#V2" target="_self">with_infima</a> )
 
<b>by </b><i><a class="ref" href="lattice3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E2:65_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> #) <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K3" target="_self">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="lattice3.html#K3" target="_self">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font></span>)</span>;<br/>
<a NAME="E3:65_1_1"/>
 <a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font> is <a href="lattice3.html#V4" target="_self">complete</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D18" target="_self">LATTICE3:def 18</a><br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font><b> such that </b><br/><a NAME="E2:65_1_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">X</font> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Maroon">a</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>  st <font color="Maroon">X</font> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Olive">b</font> holds <br/><font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" target="_self">LattPOSet</a> <span class="p2">(<span class="default"><a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>take </b><font color="Maroon">p</font> =  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">a'</font>;<br/>

<a NAME="E4:65_1_1_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">p</font> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> )
 
;<br/>
<b>thus </b><a NAME="E5:65_1_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self">LATTICE3:def 17</a><br/>

<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:65_1_1_1_1"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:65_1_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T2" target="_self">Th2</a>, </i>;<br/>
<a NAME="E4:65_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">b</font>,<font color="Maroon">a</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E5:65_1_1_1_1"/><b>then </b>
<font color="Maroon">q</font> <a href="lattice3.html#K4" target="_self">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">a'</font>
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:65_1_1_1_1"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a>, </i>;<br/>


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

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>assume </b><a NAME="E6:65_1_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E7:65_1_1_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Maroon">q</font> <a href="lattice3.html#K4" target="_self">%</a> 
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">b</font> = <font color="Maroon">q</font> <a href="lattice3.html#K4" target="_self">%</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<a NAME="E9:65_1_1_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2" target="_self">is_&lt;=_than</a> <font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">A</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" target="_self">LATTICE3:def 9</a><br/>

<b>reconsider </b><font color="Maroon">c'</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" target="_self">LattPOSet</a> <span class="p2">(<span class="default"><a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font></span>)</span></span>)</span> <b>by </b><i/>;<br/>
<b>assume </b><a NAME="E2:65_1_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E3:65_1_1_1_2"/><b>then </b>
<font color="Maroon">c'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font> <a href="lattice3.html#K4" target="_self">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T4" target="_self">Th4</a>, </i>;<br/>
<a NAME="E4:65_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">A</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font> #)
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1_1_1_2"/>
<font color="Maroon">c</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:65_1_1_1"/><b>then </b>
<font color="Maroon">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E11:65_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">A</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E12:65_1_1_1"/><b>then </b>
<font color="Maroon">a'</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font> <a href="lattice3.html#K4" target="_self">%</a> 
 
<b>by </b><i>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<b>hence </b><a NAME="E13:65_1_1_1"/>
<font color="Maroon">p</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:65_1_1"/>
 <a href="lattice3.html#K14" target="_self">latt</a> <font color="Maroon">A</font> is <a href="lattice3.html#V4" target="_self">complete</a>
 ;<br/>


</div>
