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

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<i><font color="Green" title="E47">A1</font></i>:  <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a> </span>)</span></span>)</span> <a href="filter_2.html#K24" title="FILTER_2:func.24">.:</a> 
<b>by </b><i><a class="ref" href="filter_2.html#T71" target="_self" title="FILTER_2:th.71">Th71</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>,<span class="p2"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span></span>)</span> <a href="filter_2.html#K24" title="FILTER_2:func.24">.:</a> 
<b>by </b><i><a class="ref" href="filter_2.html#T30" target="_self" title="FILTER_2:th.30">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p2"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> 
<b>by </b><i><a class="ref" href="filter_2.html#T70" target="_self" title="FILTER_2:th.70">Th70</a></i>
;<br/>
<b>assume </b><a NAME="E2:112"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 ;<br/>

<a NAME="E3:112"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="lattice2.html#T63" title="LATTICE2:th.63">LATTICE2:63</a></i>;<br/>
<a NAME="E4:112"/><b>then </b><i><font color="Green" title="E50">A4</font></i>: 
 <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 
<b>by </b><i><a class="ref" href="filter_0.html#T66" title="FILTER_0:th.66">FILTER_0:66</a></i>;<br/>
<b>hence </b><a NAME="E5:112"/>
 <a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:112"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="lattice2.html#T64" title="LATTICE2:th.64">LATTICE2:64</a></i>;<br/>

<a NAME="E6:112"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p2"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:112"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="filter_0.html#T72" title="FILTER_0:th.72">FILTER_0:72</a></i>;<br/>
<b>hence </b> <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K18" title="FILTER_2:func.18">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K18" title="FILTER_2:func.18">.&gt;</a></span></span>)</span> = 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>
<b>by </b><i><a class="txt" href="filter_2.html#E1:112"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="filter_2.html#E4:112"><i><font color="Green" title="E50">A4</font></i></a>, <a class="ref" href="lattice2.html#T79" title="LATTICE2:th.79">LATTICE2:79</a></i>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="filter_2.html#E2:112"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="lattice2.html#T78" title="LATTICE2:th.78">LATTICE2:78</a></i>
;<br/><br/>


</div>
