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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">antisymmetric</a> <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</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>thus </b><a NAME="E1:5"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a> implies <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
(  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="yellow_0.html#T23" title="YELLOW_0:th.23">YELLOW_0:23</a>, <a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K11" title="LATTICE3:func.11">"/\"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:5"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 ;<br/>

<b>assume </b><a NAME="E3:5"/>
( <font color="Maroon" title="c1">L</font> is <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="orders_2.html#V3" title="ORDERS_2:attr.3">transitive</a> )
 ;<br/>

<a NAME="E4:5"/><b>then </b><i><font color="Green" title="E6">A2</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="orders_2.html#T24" title="ORDERS_2:th.24">ORDERS_2:24</a></i>;<br/>
<a NAME="E5:5"/>
 <a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T44" title="YELLOW_0:th.44">YELLOW_0:44</a></i>;<br/>
<a NAME="E6:5"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c2">x</font> &amp; <span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="txt" href="waybel_1.html#E2:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="waybel_1.html#E4:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="yellow_0.html#T22" title="YELLOW_0:th.22">YELLOW_0:22</a></i>;<br/>
<b>hence </b><a NAME="E7:5"/>
<span class="p1">(<span class="default"><a href="yellow_0.html#K3" title="YELLOW_0:func.3">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K10" title="LATTICE3:func.10">"\/"</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font>
 <b>by </b><i><a class="ref" href="orders_2.html#T25" title="ORDERS_2:th.25">ORDERS_2:25</a></i>;<br/>


</div>
