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

<b>set </b><font color="Maroon" title="c2">R</font> =  <a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c3">S</font> =  <a href="robbins3.html#G3" title="ROBBINS3:aggr.3">LattRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> #);<br/>
<a NAME="E1:61_1_1"/>
 <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="robbins3.html#G3" title="ROBBINS3:aggr.3">LattRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> #),the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <a href="robbins3.html#G3" title="ROBBINS3:aggr.3">LattRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> #),the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <a href="robbins3.html#G3" title="ROBBINS3:aggr.3">LattRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> #) #)
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">S</font> =  <a href="robbins3.html#G3" title="ROBBINS3:aggr.3">LattRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> #) as    <a href="robbins3.html#M2" title="ROBBINS3:mode.2">LatAugmentation</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="robbins3.html#D9" target="_self" title="ROBBINS3:def.9">Def9</a></i>;<br/>
<a NAME="E3:61_1_1"/>
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 <font color="Maroon" title="c4">S</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> iff <font color="Olive" title="b1">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S</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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> iff <font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> implies <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:61_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">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">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font></span><br/><a NAME="E2:61_1_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E3:61_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/><a NAME="E4:61_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">x'</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c8">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/><b>hence </b><a NAME="E5:61_1_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 ; <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>assume </b><a NAME="E3:61_1_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font></span><br/>

<a NAME="E4:61_1_1_1"/>
<font color="Maroon" title="c7">x'</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c8">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK1" title="ROBBINS3:NK.1">|_|</a> <font color="Maroon" title="c6">y</font>
 
;<br/>
<a NAME="E5:61_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y'</font>
 
<b>by </b><i><a class="txt" href="#E3:61_1_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>
<a NAME="E6:61_1_1_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">x'</font>,<font color="Maroon" title="c8">y'</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/>
<b>hence </b><a NAME="E7:61_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</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>
<a NAME="E4:61_1_1"/><b>then </b><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c4">S</font> is <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a>
 
<b>by </b><i><a class="ref" href="robbins3.html#D10" target="_self" title="ROBBINS3:def.10">Def10</a></i>;<br/>
<a NAME="E5:61_1_1"/>
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 <font color="Maroon" title="c4">S</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">y</font> iff <font color="Olive" title="b1">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">S</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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> iff <font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c7">x'</font> = <font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c8">y'</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies <font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:61_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">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">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font></span><br/><a NAME="E2:61_1_1_2_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">S</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/><a NAME="E3:61_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y'</font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/><a NAME="E4:61_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c7">x'</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c8">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x'</font>
 <b>by </b><i><a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/><b>hence </b><a NAME="E5:61_1_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font>
 ; <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>assume </b><a NAME="E3:61_1_1_2"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</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">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font></span><br/>

<a NAME="E4:61_1_1_2"/>
<font color="Maroon" title="c7">x'</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c8">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c6">y</font>
 
;<br/>
<a NAME="E5:61_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">y'</font>
 
<b>by </b><i><a class="txt" href="#E3:61_1_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/>
<a NAME="E6:61_1_1_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">x'</font>,<font color="Maroon" title="c8">y'</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="robbins3.html#K2" title="ROBBINS3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/>
<b>hence </b><a NAME="E7:61_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</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>
<a NAME="E6:61_1_1"/><b>then </b>
<font color="Maroon" title="c4">S</font> is <a href="robbins3.html#V11" title="ROBBINS3:attr.11">naturally_inf-generated</a>
 
<b>by </b><i><a class="ref" href="robbins3.html#D11" target="_self" title="ROBBINS3:def.11">Def11</a></i>;<br/>
<b>hence </b><a NAME="E7:61_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="robbins3.html#M2" title="ROBBINS3:mode.2">LatAugmentation</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="robbins3.html#V11" title="ROBBINS3:attr.11">naturally_inf-generated</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="lattices.html#V10" title="LATTICES:attr.10">Lattice-like</a> )
 <b>by </b><i><a class="txt" href="#E4:61_1_1"><i><font color="Green" title="E27">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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