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

<b>let </b><font color="Maroon">c<sub>1</sub></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">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>





<b>assume </b><a NAME="E1:82"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<a NAME="E2:82"/>
<font color="Maroon">c<sub>2</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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="E1:82_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:82_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#R4" target="_self">is_less_than</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:82"><i><font color="Green">E48</font></i></a>, <a class="ref" href="lattice3.html#D21" target="_self">Def21</a></i>;<br/>
<b>hence </b><a NAME="E3:82_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#R1">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D17" target="_self">Def17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:82"/>
 <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K15" target="_self">"\/"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D21" target="_self">Def21</a></i>;<br/>

<a NAME="E4:82"/>
 <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></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="E1:82_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:82_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattice3.html#R3" target="_self">is_less_than</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="lattice3.html#E1:82"><i><font color="Green">E48</font></i></a>, <a class="ref" href="lattice3.html#T34" target="_self">Th34</a></i>;<br/>
<b>hence </b><a NAME="E3:82_2"/>
 <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R1">[=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#D16" target="_self">Def16</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:82"/>
 <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="lattices.html#R3">[=</a>  <a href="lattice3.html#K16" target="_self">"/\"</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lattice3.html#T34" target="_self">Th34</a></i>;<br/>


</div>
