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<b>let </b><font color="Maroon">L</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_8.html#V2">algebraic</a> <a href="waybel_0.html#NM8">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">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:38_1"/>
not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:38_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> is <a href="waybel16.html#V1" target="_self">completely-irreducible</a>
 <b>and </b><br/><a NAME="E4:38_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E5:38_1"/><i><font color="Green">E28</font></i>: 
not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><b>take </b><font color="Maroon">p</font> = <font color="Maroon">p</font>;<br/><b>thus </b><a NAME="E6:38_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="waybel16.html#K1" target="_self">Irr</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel16.html#T3" target="_self">Th3</a>, </i>;<br/><b>thus </b><a NAME="E7:38_1"/>
<font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E3:38_1"><i><font color="Green">E25</font></i></a></i>;<br/><b>thus </b><a NAME="E8:38_1"/>
not <font color="Maroon">y</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="waybel16.html#E4:38_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:38"/>
 <a href="waybel16.html#K1" target="_self">Irr</a> <font color="Maroon">L</font> is <a href="waybel_6.html#V4">order-generating</a>
 <b>by </b><i><a class="ref" href="waybel_6.html#T17">WAYBEL_6:17</a></i>;<br/>

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>

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

<b>hence </b><a NAME="E4:38"/>
 <a href="waybel16.html#K1" target="_self">Irr</a> <font color="Maroon">L</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i/>;<br/>


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