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

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> holds <br/>(  <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span> is  <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> implies (  <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span> is  <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:114"/><span class="lab"><font color="Green" title="E47">A1</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span> is  <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>  &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )</span><br/>

<a NAME="E2:114"/><span class="lab"><font color="Green" title="E48">A2</font></span>: 
<font color="Maroon">H<sub>1</sub></font>( <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T73" target="_self" title="FILTER_2:th.73">Th73</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">p9</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c5">q9</font> = <font color="Maroon" title="c3">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E49">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">a9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">q9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></span>;<br/><a NAME="E2:114_1"/><span class="lab"><font color="Green" title="E50">A4</font></span>: 
<font color="Maroon" title="c7">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c5">q9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">a9</font> = 
<font color="Maroon" title="c3">q</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></span>
<br/>.= 
<font color="Maroon" title="c5">q9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_1"><span class="lab"><font color="Green" title="E50">A4</font></span></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</font></span><br/><span class="kw">hence </span><a NAME="E4:114_1"/>
<font color="Maroon" title="c6">a9</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E50">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="114_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">a9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font> &amp; <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></span>;<br/><a NAME="E2:114_2"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E47">A1</font></span></a>, <a class="txt" href="#E2:114"><span class="lab"><font color="Green" title="E48">A2</font></span></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></span>;<br/><span class="kw">thus </span><font color="Maroon" title="c4">p9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a9</font> = 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></span>
<br/>.= 
<font color="Maroon" title="c4">p9</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:114_2"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font></span><br/><span class="kw">hence </span><a NAME="E4:114_2"/>
<font color="Maroon" title="c6">a9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E6:114"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>: 
 <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span> is   <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>   <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a>, <a class="ref" href="lattices.html#D14" title="LATTICES:def.14">LATTICES:def 14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:114"/>
 <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span> is  <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )</span><br/>

<span class="kw">thus </span><a NAME="E8:114"/>
(  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K14" title="FILTER_2:func.14">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K14" title="FILTER_2:func.14">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:114"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="txt" href="#E4:114"><span class="lab"><font color="Green" title="E49">A3</font></span></a>, <a class="txt" href="#E6:114"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="ref" href="lattices.html#D16" title="LATTICES:def.16">LATTICES:def 16</a>, <a class="ref" href="lattices.html#D17" title="LATTICES:def.17">LATTICES:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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