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

<b>set </b><font color="Maroon">R</font> =  <a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="robbins3.html#G4" target="_self">OrthoLattRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L</font> #);<br/>
<a NAME="E1:82_1_1"/>
 <a href="robbins1.html#G3">OrthoLattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L</font> #) <a href="hidden.html#R1">=</a>  <a href="robbins1.html#G3">OrthoLattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <a href="robbins3.html#G4" target="_self">OrthoLattRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L</font> #),the <a href="lattices.html#U2">L_join</a> of <a href="robbins3.html#G4" target="_self">OrthoLattRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L</font> #),the <a href="lattices.html#U1">L_meet</a> of <a href="robbins3.html#G4" target="_self">OrthoLattRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L</font> #),the <a href="robbins1.html#U1">Compl</a> of <a href="robbins3.html#G4" target="_self">OrthoLattRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L</font> #) #)
 
;<br/>
<b>then reconsider </b><font color="Maroon">S</font> =  <a href="robbins3.html#G4" target="_self">OrthoLattRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font></span>)</span>,the <a href="robbins1.html#U1">Compl</a> of <font color="Maroon">L</font> #) as    <a href="robbins3.html#M3" target="_self">CLatAugmentation</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="robbins3.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:82_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> iff <font color="Olive">x</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:82_1_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:82_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E3:82_1_1_1_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E4:82_1_1_1_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/><b>hence </b><a NAME="E5:82_1_1_1_1"/>
<font color="Maroon">x</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:82_1_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E4:82_1_1_1"/>
<font color="Maroon">x'</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E5:82_1_1_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>;<br/>
<a NAME="E6:82_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<b>hence </b><a NAME="E7:82_1_1_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</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="E4:82_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">S</font> is <a href="robbins3.html#V10" target="_self">naturally_sup-generated</a>
 
<b>by </b><i><a class="ref" href="robbins3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E5:82_1_1"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">y</font> iff <font color="Olive">x</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font>, <font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:82_1_1_2_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:82_1_1_2_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E3:82_1_1_2_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/><a NAME="E4:82_1_1_2_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x'</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/><b>hence </b><a NAME="E5:82_1_1_2_1"/>
<font color="Maroon">x</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E3:82_1_1_2"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E4:82_1_1_2"/>
<font color="Maroon">x'</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Maroon">y'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="robbins3.html#NK2" target="_self">|^|</a> <font color="Maroon">y</font>
 
;<br/>
<a NAME="E5:82_1_1_2"/><b>then </b>
<font color="Maroon">x'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">y'</font>
 
<b>by </b><i><a class="ref" href="robbins3.html#T5" target="_self">Th5</a>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>;<br/>
<a NAME="E6:82_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9">[</a><span class="default"><font color="Maroon">x'</font>,<font color="Maroon">y'</font></span><a href="filter_1.html#K9">]</a></span> <a href="hidden.html#R2">in</a>  <a href="robbins3.html#K2" target="_self">LattRel</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32">FILTER_1:32</a></i>;<br/>
<b>hence </b><a NAME="E7:82_1_1_2"/>
<font color="Maroon">x</font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">y</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="E6:82_1_1"/><b>then </b>
<font color="Maroon">S</font> is <a href="robbins3.html#V11" target="_self">naturally_inf-generated</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:82_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="robbins3.html#M3" target="_self">CLatAugmentation</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="robbins3.html#V10" target="_self">naturally_sup-generated</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="robbins3.html#V11" target="_self">naturally_inf-generated</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V10">Lattice-like</a> )
 <b>by </b><i/>;<br/>


</div>
