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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a>  <a href="robbins3.html#L3" target="_self">LattRelStr</a> ;<br/>

<b>assume </b><a NAME="E1:67_1_1"/>
<font color="Maroon">L</font> is <a href="realset2.html#V3">trivial</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L'</font> = <font color="Maroon">L</font> as  non <a href="struct_0.html#V3">empty</a> <a href="realset2.html#V3">trivial</a> <a href="orders_2.html#V2">reflexive</a>  <a href="robbins3.html#L3" target="_self">LattRelStr</a>  ;<br/>
<a NAME="E3:67_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">L'</font> is <a href="robbins3.html#V10" target="_self">naturally_sup-generated</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:67_1_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">L'</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R3">&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> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:67_1_1_1"/>
<font color="Maroon">L'</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/>


</div><b>end;</b></div>
<a NAME="E4:67_1_1"/>
<font color="Maroon">L'</font> is <a href="robbins3.html#V11" target="_self">naturally_inf-generated</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:67_1_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L'</font> holds <br/> ( <font color="Olive">x</font> <a href="orders_2.html#R3">&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> )
 
<b>by </b><i><a class="ref" href="realset2.html#D7">REALSET2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E2:67_1_1_2"/>
<font color="Maroon">L'</font> is <a href="robbins3.html#V11" target="_self">naturally_inf-generated</a>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:67_1_1"/>
( <font color="Maroon">L</font> is <a href="robbins3.html#V10" target="_self">naturally_sup-generated</a> &amp; <font color="Maroon">L</font> is <a href="robbins3.html#V11" target="_self">naturally_inf-generated</a> )
 <b>by </b><i/>;<br/>


</div>
