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

<b>let </b><font color="Maroon">C</font> be  <a href="lattice3.html#V4" target="_self">complete</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>set </b><font color="Maroon">Y</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ;<br/>
<a NAME="E1:69"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span>  <a href="lattice3.html#R4" target="_self">is_less_than</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D17" target="_self">LATTICE3:def 17</a><br/>

<b>assume </b><a NAME="E1:69_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font><b> such that </b><br/><a NAME="E3:69_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">b</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>
<a NAME="E4:69_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4" target="_self">is_less_than</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:69_1"/><b>then </b>
<font color="Maroon">b</font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a>, </i>;<br/>
<b>hence </b><a NAME="E6:69_1"/>
<font color="Maroon">c</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattice3.html#T3" target="_self">Th3</a>, <a class="ref" href="lattices.html#T27">LATTICES:27</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:69"/>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">b</font></span>)</span> where <font color="Olive">b</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">C</font> : <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> </span>}</span> ,<font color="Maroon">C</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">C</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
