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<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="lattice3.html#V4" title="LATTICE3:attr.4">complete</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>



<a NAME="E1:42"/>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a>  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">LATTICE3:def 21</a></i>;<br/>
<a NAME="E2:42"/><b>then </b><i><font color="Green" title="E28">A1</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <span class="p1">(<span class="default"><a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T30" title="LATTICE3:th.30">LATTICE3:30</a></i>;<br/>
<div><i><font color="Green" title="E29">A2</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" title="c3">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:42_1"/>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c3">r</font>
 ;<br/><a NAME="E2:42_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#R4" title="LATTICE3:pred.4">is_less_than</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T31" title="LATTICE3:th.31">LATTICE3:31</a></i>;<br/><a NAME="E3:42_1"/><b>then </b>
(  <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#D21" title="LATTICE3:def.21">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" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:42"/><b>then </b>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c2">X</font>, <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E2:42"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="yellow_0.html#T15" target="_self" title="YELLOW_0:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E5:42"/>
 <a href="lattice3.html#K15" title="LATTICE3:func.15">"\/"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_0.html#E2:42"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="yellow_0.html#E3:42"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="yellow_0.html#D9" target="_self" title="YELLOW_0:def.9">Def9</a></i>;<br/>

<a NAME="E6:42"/>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/>
<a NAME="E7:42"/><b>then </b><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a> 
 
<b>by </b><i><a class="ref" href="lattice3.html#T28" title="LATTICE3:th.28">LATTICE3:28</a></i>;<br/>
<div><i><font color="Green" title="E31">A4</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" title="c3">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/><b>assume </b><a NAME="E1:42_2"/>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#NR3" title="LATTICE3:NR.3">is_&gt;=_than</a> <font color="Maroon" title="c3">r</font>
 ;<br/><a NAME="E2:42_2"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattice3.html#NR7" title="LATTICE3:NR.7">is_greater_than</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T29" title="LATTICE3:th.29">LATTICE3:29</a></i>;<br/><a NAME="E3:42_2"/><b>then </b>
(  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a>  <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> &amp; <span class="p1">(<span class="default"><a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font> &amp;  <a href="lattice3.html#K5" title="LATTICE3:func.5">%</a> <font color="Maroon" title="c3">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">r</font> )
 <b>by </b><i><a class="ref" href="lattice3.html#T34" title="LATTICE3:th.34">LATTICE3:34</a></i>;<br/><b>hence </b><a NAME="E4:42_2"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font></span>)</span> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#NR2" title="ORDERS_2:NR.2">&gt;=</a> <font color="Maroon" title="c3">r</font>
 <b>by </b><i><a class="ref" href="lattice3.html#T7" title="LATTICE3:th.7">LATTICE3:7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:42"/><b>then </b>
 <a href="yellow_0.html#R2" title="YELLOW_0:pred.2">ex_inf_of</a> <font color="Maroon" title="c2">X</font>, <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="txt" href="yellow_0.html#E7:42"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="yellow_0.html#T16" target="_self" title="YELLOW_0:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E10:42"/>
 <a href="lattice3.html#K16" title="LATTICE3:func.16">"/\"</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#K2" title="YELLOW_0:func.2">"/\"</a> <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow_0.html#E7:42"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="yellow_0.html#E8:42"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="yellow_0.html#D10" target="_self" title="YELLOW_0:def.10">Def10</a></i>;<br/>


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