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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_8.html#V2" title="WAYBEL_8:attr.2">algebraic</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/><b>assume </b><a NAME="E1:38_1"/>
not <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c2">x</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green" title="E24">A1</font></i>: 
<font color="Maroon" title="c4">p</font> is <a href="waybel16.html#V1" title="WAYBEL16:attr.1">completely-irreducible</a>
 <b>and </b><br/><a NAME="E4:38_1"/><i><font color="Green" title="E25">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">p</font>
 <b>and </b><br/><a NAME="E5:38_1"/><i><font color="Green" title="E26">A3</font></i>: 
not <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">p</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T31" target="_self" title="WAYBEL16:th.31">Th31</a></i>;<br/><b>take </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c4">p</font>;<br/><b>thus </b><a NAME="E6:38_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel16.html#K1" title="WAYBEL16:func.1">Irr</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E3:38_1"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="waybel16.html#D4" target="_self" title="WAYBEL16:def.4">Def4</a></i>;<br/><b>thus </b><a NAME="E7:38_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E4:38_1"><i><font color="Green" title="E25">A2</font></i></a></i>;<br/><b>thus </b><a NAME="E8:38_1"/>
not <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E5:38_1"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:38"/>
 <a href="waybel16.html#K1" title="WAYBEL16:func.1">Irr</a> <font color="Maroon" title="c1">L</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#T17" title="WAYBEL_6:th.17">WAYBEL_6:17</a></i>;<br/>

<b>let </b><font color="Maroon" title="c2">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E3:38"/>
<font color="Maroon" title="c2">X</font> is <a href="waybel_6.html#V4" title="WAYBEL_6:attr.4">order-generating</a>
 ;<br/>

<b>hence </b><a NAME="E4:38"/>
 <a href="waybel16.html#K1" title="WAYBEL16:func.1">Irr</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T25" target="_self" title="WAYBEL16:th.25">Th25</a></i>;<br/>


</div>
