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

<b>let </b><font color="Maroon">P</font> be  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V2">upper-bounded</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:45"/><i><font color="Green">E37</font></i>: 
the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">P</font> is <a href="wellord1.html#V2">well-ordering</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">P</font> as  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#V16">connected</a> <a href="waybel_3.html#V3">continuous</a> <a href="orders_2.html#NM1">Poset</a> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_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="E1:45_1"/>
<font color="Maroon">x</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">y</font>
 ;<br/><a NAME="E2:45_1"/><b>then </b>
( ( <font color="Maroon">x</font> is <a href="waybel_3.html#V1">compact</a> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> ) or <font color="Maroon">x</font> <a href="orders_2.html#R2">&lt;</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="waybel_3.html#T1">WAYBEL_3:1</a>, <a class="ref" href="waybel_3.html#T14">WAYBEL_3:14</a></i>;<br/><b>then consider </b><font color="Maroon">z</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:45_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">z</font> is <a href="waybel_3.html#V1">compact</a> &amp; <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">z</font> = <font color="Maroon">z</font>;<br/><b>thus </b><a NAME="E5:45_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel_8.html#K1">CompactSublatt</a> <font color="Maroon">L</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="waybel_8.html#D1">WAYBEL_8:def 1</a></i>;<br/><b>thus </b><a NAME="E6:45_1"/>
( <font color="Maroon">x</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">z</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:45"/>
<font color="Maroon">P</font> is <a href="waybel_8.html#V2">algebraic</a>
 <b>by </b><i><a class="ref" href="waybel_8.html#T7">WAYBEL_8:7</a></i>;<br/>


</div>
