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

<b>let </b><font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c2">L2</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon" title="c1">L1</font> is <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a> &amp; <font color="Maroon" title="c2">L2</font> is <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a> implies <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> is <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:67_1"/><i><font color="Green" title="E44">A1</font></i>: 
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  ex <font color="Olive" title="b3">r1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> &amp; ( for <font color="Olive" title="b4">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> holds <br/><font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">r1</font> ) )
 ; <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>assume </b><a NAME="E2:67_1"/><i><font color="Green" title="E45">A2</font></i>: 
for <font color="Olive" title="b1">p2</font>, <font color="Olive" title="b2">q2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  ex <font color="Olive" title="b3">r2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q2</font> &amp; ( for <font color="Olive" title="b4">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Olive" title="b1">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q2</font> holds <br/><font color="Olive" title="b4">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">r2</font> ) )
 ; <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>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></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>consider </b><font color="Maroon" title="c5">p1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c6">p2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E4:67_1"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c8">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E6:67_1"/><i><font color="Green" title="E47">A4</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c7">q1</font>,<font color="Maroon" title="c8">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font><b> such that </b><br/><a NAME="E8:67_1"/><i><font color="Green" title="E48">A5</font></i>: 
( <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font> &amp; ( for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font> holds <br/><font color="Olive" title="b1">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r1</font> ) )
 <b>by </b><i><a class="txt" href="filter_1.html#E1:67_1"><i><font color="Green" title="E44">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E10:67_1"/><i><font color="Green" title="E49">A6</font></i>: 
( <font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q2</font> &amp; ( for <font color="Olive" title="b1">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>  st <font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q2</font> holds <br/><font color="Olive" title="b1">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">r2</font> ) )
 <b>by </b><i><a class="txt" href="filter_1.html#E2:67_1"><i><font color="Green" title="E45">A2</font></i></a></i>;<br/>
<b>take </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
;<br/>

<a NAME="E11:67_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E4:67_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E12:67_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E6:67_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="txt" href="filter_1.html#E8:67_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="filter_1.html#E10:67_1"><i><font color="Green" title="E49">A6</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>

<b>let </b><font color="Maroon" title="c11">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>;<br/>

<b>consider </b><font color="Maroon" title="c12">s1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c13">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E14:67_1"/><i><font color="Green" title="E50">A7</font></i>: 
<font color="Maroon" title="c11">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c12">s1</font>,<font color="Maroon" title="c13">s2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>assume </b><a NAME="E15:67_1"/>
<font color="Maroon" title="c3">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 ;<br/>

<a NAME="E16:67_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">s1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E4:67_1"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="filter_1.html#E14:67_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th36</a></i>;<br/>
<a NAME="E17:67_1"/><b>then </b>
( <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font> &amp; <font color="Maroon" title="c6">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">q2</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E6:67_1"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>
<a NAME="E18:67_1"/><b>then </b>
( <font color="Maroon" title="c12">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r1</font> &amp; <font color="Maroon" title="c13">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c10">r2</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E8:67_1"><i><font color="Green" title="E48">A5</font></i></a>, <a class="txt" href="filter_1.html#E10:67_1"><i><font color="Green" title="E49">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E19:67_1"/>
<font color="Maroon" title="c11">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c9">r1</font>,<font color="Maroon" title="c10">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="txt" href="filter_1.html#E14:67_1"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:67"/><i><font color="Green" title="E44">A8</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  ex <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span> st <br/>( <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> &amp; ( for <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <font color="Olive" title="b1">a</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b4">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">c</font> ) )
 ; <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>thus </b><a NAME="E3:67"/>
for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">q1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  ex <font color="Olive" title="b3">r1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font> st <br/>( <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> &amp; ( for <font color="Olive" title="b4">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>  st <font color="Olive" title="b1">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q1</font> holds <br/><font color="Olive" title="b4">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">r1</font> ) )
  <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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">q1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>;<br/>



<b>consider </b><font color="Maroon" title="c5">p2</font>, <font color="Maroon" title="c6">q2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span><b> such that </b><br/><a NAME="E3:67_2"/><i><font color="Green" title="E45">A9</font></i>: 
( <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> holds <br/><font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font> ) )
 <b>by </b><i><a class="txt" href="filter_1.html#E2:67"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c9">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E5:67_2"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">r1</font>
;<br/>

<a NAME="E6:67_2"/><i><font color="Green" title="E47">A11</font></i>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E5:67_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th36</a></i>;<br/>
<a NAME="E7:67_2"/><b>then </b><i><font color="Green" title="E48">A12</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q1</font> &amp; <font color="Maroon" title="c5">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q2</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:67_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>
<b>thus </b><a NAME="E8:67_2"/>
<font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q1</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E3:67_2"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="filter_1.html#E6:67_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>

<b>let </b><font color="Maroon" title="c10">s1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>;<br/>

<b>assume </b><a NAME="E9:67_2"/>
<font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q1</font>
 ;<br/>

<a NAME="E10:67_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E7:67_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>
<a NAME="E11:67_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c6">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<b>by </b><i><a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th36</a></i>;<br/>
<a NAME="E12:67_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c10">s1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E3:67_2"><i><font color="Green" title="E45">A9</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:67_2"/>
<font color="Maroon" title="c10">s1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">r1</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E5:67_2"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">p2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>; <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>let </b><font color="Maroon" title="c4">q2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/>



<b>consider </b><font color="Maroon" title="c5">p1</font>, <font color="Maroon" title="c6">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">c</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span><b> such that </b><br/><a NAME="E6:67"/><i><font color="Green" title="E45">A13</font></i>: 
( <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> &amp; ( for <font color="Olive" title="b1">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1"><a href="filter_1.html#K7" title="FILTER_1:func.7">[:</a><span class="default"><font color="Maroon" title="c1">L1</font>,<font color="Maroon" title="c2">L2</font></span><a href="filter_1.html#K7" title="FILTER_1:func.7">:]</a></span>  st <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> holds <br/><font color="Olive" title="b1">d</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font> ) )
 <b>by </b><i><a class="txt" href="filter_1.html#E2:67"><i><font color="Green" title="E44">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">r1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L1</font>, <font color="Maroon" title="c9">r2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font><b> such that </b><br/><a NAME="E8:67"/><i><font color="Green" title="E46">A14</font></i>: 
<font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c9">r2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c9">r2</font>
;<br/>

<a NAME="E9:67"/><i><font color="Green" title="E47">A15</font></i>: 
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E8:67"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th36</a></i>;<br/>
<a NAME="E10:67"/><b>then </b><i><font color="Green" title="E48">A16</font></i>: 
( <font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q1</font> &amp; <font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q2</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E6:67"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>
<b>thus </b><a NAME="E11:67"/>
<font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">r2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q2</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E6:67"><i><font color="Green" title="E45">A13</font></i></a>, <a class="txt" href="filter_1.html#E9:67"><i><font color="Green" title="E47">A15</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>

<b>let </b><font color="Maroon" title="c10">s2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L2</font>;<br/>

<b>assume </b><a NAME="E12:67"/>
<font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q2</font>
 ;<br/>

<a NAME="E13:67"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p2</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">s2</font></span>)</span></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<b>by </b><i><a class="txt" href="filter_1.html#E10:67"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>
<a NAME="E14:67"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c3">p2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c10">s2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c4">q2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span>
 
<b>by </b><i><a class="ref" href="filter_1.html#T36" target="_self" title="FILTER_1:th.36">Th36</a></i>;<br/>
<a NAME="E15:67"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c8">r1</font>,<font color="Maroon" title="c10">s2</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">c</font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E6:67"><i><font color="Green" title="E45">A13</font></i></a></i>;<br/>
<b>hence </b><a NAME="E16:67"/>
<font color="Maroon" title="c10">s2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">r2</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E8:67"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="filter_1.html#T37" target="_self" title="FILTER_1:th.37">Th37</a></i>;<br/>


</div>
