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



<a NAME="E1:42"/>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a>  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E2:42"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T30">LATTICE3:30</a></i>;<br/>
<div><i><font color="Green">E29</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon">X</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:42_1"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#R4">is_less_than</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T31">LATTICE3:31</a></i>;<br/><a NAME="E3:42_1"/><b>then </b>
(  <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="lattices.html#R1">[=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">r</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D21">LATTICE3:def 21</a></i>;<br/><b>hence </b><a NAME="E4:42_1"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:42"/><b>then </b>
 <a href="yellow_0.html#R1" target="_self">ex_sup_of</a> <font color="Maroon">X</font>, <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:42"/><b>then </b>
 <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i>, <a class="txt" href="yellow_0.html#E3:42"><i><font color="Green">E29</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E6:42"/>
 <a href="lattice3.html#K15">"\/"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1" target="_self">"\/"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>

<a NAME="E7:42"/>
<font color="Maroon">X</font> <a href="lattice3.html#NR7" target="_self">is_great_than</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/>
<a NAME="E8:42"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T28">LATTICE3:28</a></i>;<br/>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:42_2"/>
<font color="Maroon">X</font> <a href="lattice3.html#NR3" target="_self">is_&gt;=_than</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E2:42_2"/><b>then </b>
<font color="Maroon">X</font> <a href="lattice3.html#NR7" target="_self">is_great_than</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T29">LATTICE3:29</a></i>;<br/><a NAME="E3:42_2"/><b>then </b>
(  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font> <a href="lattices.html#R1">[=</a>  <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <font color="Maroon">r</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font> &amp;  <a href="lattice3.html#K5">%</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T34">LATTICE3:34</a></i>;<br/><b>hence </b><a NAME="E4:42_2"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a>  <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:42"/><b>then </b>
 <a href="yellow_0.html#R2" target="_self">ex_inf_of</a> <font color="Maroon">X</font>, <a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E8:42"><i><font color="Green">E30</font></i></a>, </i>;<br/>
<a NAME="E11:42"/><b>then </b>
 <a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font></span>)</span> <a href="lattice3.html#K4">%</a> 
 
<b>by </b><i><a class="txt" href="yellow_0.html#E8:42"><i><font color="Green">E30</font></i></a>, <a class="txt" href="yellow_0.html#E9:42"><i><font color="Green">E31</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E12:42"/>
 <a href="lattice3.html#K16">"/\"</a> <font color="Maroon">X</font>,<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K2" target="_self">"/\"</a> <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">L</font></span>)</span>
 ;<br/>


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