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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  <a href="robbins3.html#L3" title="ROBBINS3:struct.3">LattRelStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V11" title="ROBBINS3:attr.11">naturally_inf-generated</a> ) )</span><br/>

<b>assume </b><a NAME="E1:67_1_1"/>
<font color="Maroon" title="c1">L</font> is <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V11" title="ROBBINS3:attr.11">naturally_inf-generated</a> )</span><br/>

<b>then reconsider </b><font color="Maroon" title="c2">L'</font> = <font color="Maroon" title="c1">L</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a>  <a href="robbins3.html#L3" title="ROBBINS3:struct.3">LattRelStr</a>  ;<br/>
<a NAME="E3:67_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">L'</font> is <a href="robbins3.html#V10" title="ROBBINS3:attr.10">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" 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="c2">L'</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&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> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:67_1_1_1"/>
<font color="Maroon" title="c2">L'</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>; <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="E4:67_1_1"/>
<font color="Maroon" title="c2">L'</font> is <a href="robbins3.html#V11" title="ROBBINS3:attr.11">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" 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="c2">L'</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&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> )
 
<b>by </b><i><a class="ref" href="struct_0.html#D10" title="STRUCT_0:def.10">STRUCT_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:67_1_1_2"/>
<font color="Maroon" title="c2">L'</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>; <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>hence </b><a NAME="E5:67_1_1"/>
( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V10" title="ROBBINS3:attr.10">naturally_sup-generated</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V11" title="ROBBINS3:attr.11">naturally_inf-generated</a> )
 <b>by </b><i><a class="txt" href="#E3:67_1_1"><i><font color="Green" title="E27">A1</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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