<?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>, <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>;<br/>





<b>assume </b><a NAME="E1:109"/><i><font color="Green" title="E44">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a>
 ;<br/>

<a NAME="E2:109"/><b>then </b><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>
 
;<br/>
<b>assume </b><a NAME="E3:109"/><i><font color="Green" title="E46">A3</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>
 ;<br/>

<b>set </b><font color="Maroon" title="c4">P</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>;<br/>
<b>set </b><font color="Maroon" title="c5">K</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>;<br/>
<a NAME="E4:109"/><i><font color="Green" title="E47">A4</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>let </b><font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c7">b'</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c7">b'</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="E6:109"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">a</font> &amp; <font color="Maroon" title="c9">b</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:109"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">c</font> = <font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font>;<br/>
<b>set </b><font color="Maroon" title="c11">d</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font>;<br/>
<a NAME="E7:109"/>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<a NAME="E8:109"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:109"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="filter_0.html#T7" title="FILTER_0:th.7">FILTER_0:7</a>, <a class="ref" href="lattices.html#T23" title="LATTICES:th.23">LATTICES:23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">d'</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <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#E3:109"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">d'</font>
;<br/>

<a NAME="E10:109"/><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">a</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> &amp; <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">d'</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E2:109"><i><font color="Green" title="E45">A2</font></i></a>, <a class="txt" href="filter_2.html#E6:109"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a>, <a class="ref" href="lattices.html#D12" title="LATTICES:def.12">LATTICES:def 12</a></i>;<br/>
<a NAME="E11:109"/>
<font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="filter_2.html#E1:109"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></i>;<br/>
<a NAME="E12:109"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="txt" href="filter_2.html#E6:109"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></i>;<br/>
<a NAME="E13:109"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span></span>)</span></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T2" title="FILTER_0:th.2">FILTER_0:2</a></i>;<br/>
<b>hence </b><a NAME="E14:109"/>
<font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">d'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b'</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E10:109"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="filter_2.html#T75" target="_self" title="FILTER_2:th.75">Th75</a></i>;<br/>

<b>let </b><font color="Maroon" title="c13">e'</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>;<br/>

<b>reconsider </b><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c13">e'</font>, <font color="Maroon" title="c15">ae</font> = <font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">e'</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/>
<b>assume </b><a NAME="E16:109"/>
<font color="Maroon" title="c6">a'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">e'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">b'</font>
 ;<br/>

<a NAME="E17:109"/><b>then </b>
<font color="Maroon" title="c15">ae</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font>
 
<b>by </b><i><a class="ref" href="filter_2.html#T75" target="_self" title="FILTER_2:th.75">Th75</a></i>;<br/>
<a NAME="E18:109"/><b>then </b>
( <font color="Maroon" title="c8">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c14">e</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E3:109"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a>, <a class="ref" href="filter_2.html#T74" target="_self" title="FILTER_2:th.74">Th74</a></i>;<br/>
<a NAME="E19:109"/><b>then </b>
( <font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">e</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c14">e</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#E1:109"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="filter_2.html#E3:109"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="filter_2.html#E4:109"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a>, <a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a>, <a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></i>;<br/>
<a NAME="E20:109"/><b>then </b>
( <font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">e</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T1" title="FILTER_0:th.1">FILTER_0:1</a>, <a class="ref" href="lattices.html#T21" title="LATTICES:th.21">LATTICES:21</a></i>;<br/>
<a NAME="E21:109"/><b>then </b>
<font color="Maroon" title="c14">e</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">a</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c9">b</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T27" title="LATTICES:th.27">LATTICES:27</a></i>;<br/>
<b>hence </b><a NAME="E22:109"/>
<font color="Maroon" title="c13">e'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">d'</font>
 <b>by </b><i><a class="ref" href="filter_2.html#T75" target="_self" title="FILTER_2:th.75">Th75</a></i>;<br/>


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