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

<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>





<b>assume </b><a NAME="E1:114"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 ;<br/>

<a NAME="E2:114"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span>) <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font>, <font color="Maroon">q'</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span> <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E2:114_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">p</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a'</font> = 
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p'</font>
<b>by </b><i>, <a class="ref" href="lattices.html#T21">LATTICES:21</a></i>
;<br/><b>hence </b><a NAME="E4:114_1"/>
<font color="Maroon">a'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green">E22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a'</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a'</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i/>;<br/><a NAME="E2:114_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><font color="Maroon">q'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a'</font> = 
<font color="Maroon">q</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">q'</font>
<b>by </b><i>, <a class="ref" href="lattices.html#D3">LATTICES:def 3</a></i>
;<br/><b>hence </b><a NAME="E4:114_2"/>
<font color="Maroon">a'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E6:114"/><b>then </b><i><font color="Green">E140</font></i>: 
 <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span> is  <a href="lattices.html#V13">lower-bounded</a> <a href="lattices.html#V14">upper-bounded</a> <a href="lattices.html#NM1">Lattice</a>
 
<b>by </b><i>, <a class="ref" href="lattices.html#D13">LATTICES:def 13</a>, <a class="ref" href="lattices.html#D14">LATTICES:def 14</a></i>;<br/>
<b>hence </b><a NAME="E7:114"/>
 <a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span> is <a href="lattices.html#V15">bounded</a>
 ;<br/>

<b>thus </b><a NAME="E8:114"/>
(  <a href="lattices.html#K6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K23" target="_self">latt</a> <font color="Maroon">L</font>,<span class="p2"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="filter_2.html#K22" target="_self">#]</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , , <a class="ref" href="lattices.html#D16">LATTICES:def 16</a>, <a class="ref" href="lattices.html#D17">LATTICES:def 17</a></i>;<br/>


</div>
