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

<b>let </b><font color="Maroon">L1</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">L2</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>



<b>thus </b><a NAME="E1:67"/>
( <font color="Maroon">L1</font> is <a href="filter_0.html#V3">implicative</a> &amp; <font color="Maroon">L2</font> is <a href="filter_0.html#V3">implicative</a> implies <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> is <a href="filter_0.html#V3">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">E68</font></i>: 
for <font color="Olive">p1</font>, <font color="Olive">q1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  ex <font color="Olive">r1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> st <br/>( <font color="Olive">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">r1</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q1</font> &amp; ( for <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st <font color="Olive">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">s1</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q1</font> holds <br/><font color="Olive">s1</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r1</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/>

<b>assume </b><a NAME="E2:67_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">p2</font>, <font color="Olive">q2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>  ex <font color="Olive">r2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font> st <br/>( <font color="Olive">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">r2</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q2</font> &amp; ( for <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>  st <font color="Olive">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">s2</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q2</font> holds <br/><font color="Olive">s2</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r2</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/>

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/>

<b>consider </b><font color="Maroon">p1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">p2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E4:67_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">q2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E6:67_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font><b> such that </b><br/><a NAME="E8:67_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> &amp; ( for <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st <font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">s1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> holds <br/><font color="Olive">s1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r1</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E10:67_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> &amp; ( for <font color="Olive">s2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>  st <font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">s2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> holds <br/><font color="Olive">s2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r2</font> ) )
 <b>by </b><i/>;<br/>
<b>take </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
;<br/>

<a NAME="E11:67_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E12:67_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , , </i>;<br/>

<b>let </b><font color="Maroon">d</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>;<br/>

<b>consider </b><font color="Maroon">s1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">s2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E14:67_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">s2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>assume </b><a NAME="E15:67_1"/>
<font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E16:67_1"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E17:67_1"/><b>then </b>
( <font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:67_1"/><b>then </b>
( <font color="Maroon">s1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">s2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E19:67_1"/>
<font color="Maroon">d</font> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i>, </i>;<br/>


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

<b>assume </b><a NAME="E2:67"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>  ex <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span> st <br/>( <font color="Olive">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">c</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>  st <font color="Olive">a</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Olive">b</font> holds <br/><font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Olive">c</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/>

<b>thus </b><a NAME="E3:67"/>
for <font color="Olive">p1</font>, <font color="Olive">q1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  ex <font color="Olive">r1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font> st <br/>( <font color="Olive">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">r1</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q1</font> &amp; ( for <font color="Olive">s1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>  st <font color="Olive">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">s1</font> <a href="lattices.html#R3">[=</a> <font color="Olive">q1</font> holds <br/><font color="Olive">s1</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r1</font> ) )
  <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">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">p1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/><b>let </b><font color="Maroon">q1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/>



<b>consider </b><font color="Maroon">p2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>, <font color="Maroon">q2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span><b> such that </b><br/><a NAME="E3:67_2"/><i><font color="Green">E86</font></i>: 
( <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>  st <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> holds <br/><font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">r2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E5:67_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">r1</font>
;<br/>

<a NAME="E6:67_2"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:67_2"/><b>then </b><i><font color="Green">E91</font></i>: 
( <font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E8:67_2"/>
<font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font>
 <b>by </b><i>, , </i>;<br/>

<b>let </b><font color="Maroon">s1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/>

<b>assume </b><a NAME="E9:67_2"/>
<font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font>
 ;<br/>

<a NAME="E10:67_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:67_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E12:67_2"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">s1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E13:67_2"/>
<font color="Maroon">s1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r1</font>
 <b>by </b><i>, </i>;<br/>


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

<b>let </b><font color="Maroon">p2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7">FILTER_0:def 7</a><br/><b>let </b><font color="Maroon">q2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/>



<b>consider </b><font color="Maroon">p1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">q1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>;<br/>
<b>consider </b><font color="Maroon">c</font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span><b> such that </b><br/><a NAME="E6:67"/><i><font color="Green">E92</font></i>: 
( <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> &amp; ( for <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1"><a href="filter_1.html#K7" target="_self">[:</a><span class="default"><font color="Maroon">L1</font>,<font color="Maroon">L2</font></span><a href="filter_1.html#K7" target="_self">:]</a></span>  st <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> holds <br/><font color="Olive">d</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">r1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L1</font>, <font color="Maroon">r2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font><b> such that </b><br/><a NAME="E8:67"/><i><font color="Green">E169</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">r2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9">DOMAIN_1:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">r2</font>
;<br/>

<a NAME="E9:67"/><i><font color="Green">E170</font></i>: 
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:67"/><b>then </b><i><font color="Green">E171</font></i>: 
( <font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font> )
 
<b>by </b><i>, </i>;<br/>
<b>thus </b><a NAME="E11:67"/>
<font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font>
 <b>by </b><i>, , </i>;<br/>

<b>let </b><font color="Maroon">s2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L2</font>;<br/>

<b>assume </b><a NAME="E12:67"/>
<font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q2</font>
 ;<br/>

<a NAME="E13:67"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s2</font></span>)</span></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:67"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#K4">"/\"</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">s2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="filter_1.html#K9" target="_self">]</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E15:67"/><b>then </b>
<span class="p1"><a href="filter_1.html#K9" target="_self">[</a><span class="default"><font color="Maroon">r1</font>,<font color="Maroon">s2</font></span><a href="filter_1.html#K9" target="_self">]</a></span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:67"/>
<font color="Maroon">s2</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r2</font>
 <b>by </b><i>, </i>;<br/>


</div>
