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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> ) )</span><br/>





<b>assume </b><a NAME="E1:114"/><i><font color="Green" title="E49">A1</font></i>: 
<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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <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#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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"/><i><font color="Green" title="E50">A2</font></i>: 
<font color="Maroon">H<sub>1</sub></font>( <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#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span>
 
<b>by </b><i><a class="ref" href="filter_2.html#T73" target="_self" title="FILTER_2:th.73">Th73</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c4">p'</font> = <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c5">q'</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span>)</span> <b>by </b><i><a class="txt" href="filter_2.html#E1:114"><i><font color="Green" title="E49">A1</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></i>;<br/>
<div><i><font color="Green" title="E51">A3</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" title="c6">a'</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></i>;<br/><a NAME="E2:114_1"/><i><font color="Green" title="E52">A4</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:114"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:114"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></i>;<br/><b>thus </b><font color="Maroon" title="c4">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">a'</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>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<font color="Maroon" title="c4">p'</font>
<b>by </b><i><a class="txt" href="filter_2.html#E2:114_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font></span><br/><b>hence </b><a NAME="E4:114_1"/>
<font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E52">A5</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" title="c6">a'</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">q'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q'</font> &amp; <font color="Maroon" title="c6">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q'</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c6">a'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> <b>by </b><i><a class="ref" href="filter_2.html#T69" target="_self" title="FILTER_2:th.69">Th69</a></i>;<br/><a NAME="E2:114_2"/><i><font color="Green" title="E53">A6</font></i>: 
<font color="Maroon" title="c7">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E1:114"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:114"><i><font color="Green" title="E50">A2</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></i>;<br/><b>thus </b><font color="Maroon" title="c5">q'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c6">a'</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>
<b>by </b><i><a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>
<br/>.= 
<font color="Maroon" title="c5">q'</font>
<b>by </b><i><a class="txt" href="filter_2.html#E2:114_2"><i><font color="Green" title="E53">A6</font></i></a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q'</font></span><br/><b>hence </b><a NAME="E4:114_2"/>
<font color="Maroon" title="c6">a'</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:114"/><b>then </b><i><font color="Green" title="E53">A7</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#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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>
 
<b>by </b><i><a class="txt" href="filter_2.html#E4:114"><i><font color="Green" title="E51">A3</font></i></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></i>;<br/>
<b>hence </b><a NAME="E7:114"/>
 <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#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> is <a href="lattices.html#V15" title="LATTICES:attr.15">bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )</span><br/>

<b>thus </b><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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</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#K23" title="FILTER_2:func.23">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E4:114"><i><font color="Green" title="E51">A3</font></i></a>, <a class="txt" href="filter_2.html#E5:114"><i><font color="Green" title="E52">A5</font></i></a>, <a class="txt" href="filter_2.html#E6:114"><i><font color="Green" title="E53">A7</font></i></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></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
