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

<b>let </b><font color="Maroon">L1</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">L2</font> be  <a href="yellow_0.html#V1">lower-bounded</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L1</font>,<font color="Maroon">L2</font>;<br/>



<b>assume </b><a NAME="E1:11"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> is <a href="waybel_0.html#V17">infs-preserving</a> &amp; <font color="Maroon">f</font> is <a href="waybel_0.html#V18">sups-preserving</a> )
 ;<br/>

<b>thus </b><a NAME="E2:11"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V19">meet-preserving</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1"><b>proof </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">L1</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D34">WAYBEL_0:def 34</a><br/>

<b>thus </b><a NAME="E1:11_1"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R3">preserves_inf_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D32">WAYBEL_0:def 32</a></i>;<br/>


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

<b>thus </b><a NAME="E3:11"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V20">join-preserving</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_2"><b>proof </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">L1</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D35">WAYBEL_0:def 35</a><br/>

<b>thus </b><a NAME="E1:11_2"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i>, <a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>


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


</div>
