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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>thus </b><a NAME="E1:61"/>
( <font color="Maroon">p</font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">X</font> implies <font color="Maroon">p</font> <a href="lattice3.html#K4" target="_self">%</a>  <a href="lattice3.html#R1" target="_self">is_&lt;=_than</a> <font color="Maroon">X</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:61_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self">LATTICE3:def 16</a><br/>

<b>let </b><font color="Maroon">p'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" target="_self">LattPOSet</a> <font color="Maroon">L</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" target="_self">LATTICE3:def 8</a><br/>

<a NAME="E2:61_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">p'</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">p'</font></span>)</span> <a href="lattice3.html#K4" target="_self">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">p'</font> )
 
;<br/>
<b>assume </b><a NAME="E3:61_1"/>
<font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:61_1"/><b>then </b>
<font color="Maroon">p</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K5" target="_self">%</a> <font color="Maroon">p'</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:61_1"/>
<font color="Maroon">p</font> <a href="lattice3.html#K4" target="_self">%</a>  <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">p'</font>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" target="_self">LattPOSet</a> <font color="Maroon">L</font></span>)</span>  st <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> holds <br/><font color="Maroon">p</font> <a href="lattice3.html#K4" target="_self">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">q'</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D8" target="_self">LATTICE3:def 8</a><br/>

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D16" target="_self">LATTICE3:def 16</a><br/>

<b>assume </b><a NAME="E3:61"/>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/>

<a NAME="E4:61"/><b>then </b>
<font color="Maroon">p</font> <a href="lattice3.html#K4" target="_self">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">q</font> <a href="lattice3.html#K4" target="_self">%</a> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:61"/>
<font color="Maroon">p</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/>


</div>
