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

<b>let </b><font color="Maroon">I</font> be   <a href="filter_0.html#NM2">I_Lattice</a>;<br/><b>let </b><font color="Maroon">FI</font> be    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">i</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">j</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>;<br/>







<b>set </b><font color="Maroon">R</font> =  <a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font>;<br/>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">b</font> = <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font>;<br/>
<b>thus </b><a NAME="E1:26"/>
( <font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font> implies <font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:26_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a><br/>

<a NAME="E2:26_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font> &amp;  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font> &amp; <font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span>,<font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span>,<font color="Maroon">j</font> )
 
<b>by </b><i>, , <a class="ref" href="eqrel_1.html#T28">EQREL_1:28</a></i>;<br/>
<a NAME="E3:26_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">j</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T30">EQREL_1:30</a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a>, <a class="ref" href="lattices.html#T22">LATTICES:22</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E4:26_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="filter_0.html#K4">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span></span>)</span> &amp; <font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D12">FILTER_0:def 12</a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<a NAME="E5:26_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:26_1"/>
<font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T9">FILTER_0:9</a></i>;<br/>


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

<b>assume </b><a NAME="E2:26"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font>
 ;<br/>

<a NAME="E3:26"/>
( <font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="lattices.html#K4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> &amp; <font color="Maroon">j</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">I</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T2">FILTER_0:2</a>, <a class="ref" href="filter_0.html#T3">FILTER_0:3</a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a>, <a class="ref" href="lattices.html#T17">LATTICES:17</a>, <a class="ref" href="lattices.html#D11">LATTICES:def 11</a></i>;<br/>
<a NAME="E4:26"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6">Top</a> <font color="Maroon">I</font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font> &amp; <font color="Maroon">I</font> is   <a href="lattices.html#NM5">1_Lattice</a> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T4">FILTER_0:4</a>, <a class="ref" href="filter_0.html#T37">FILTER_0:37</a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<a NAME="E5:26"/><b>then </b>
( <font color="Maroon">i</font> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> <a href="lattices.html#R3">[=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> &amp;  <a href="lattices.html#K6">Top</a> <font color="Maroon">I</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">j</font> <a href="filter_0.html#K4">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> &amp;  <a href="lattices.html#K6">Top</a> <font color="Maroon">I</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T12">FILTER_0:12</a>, <a class="ref" href="filter_0.html#D8">FILTER_0:def 8</a></i>;<br/>
<a NAME="E6:26"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font> &amp; <font color="Maroon">j</font> <a href="filter_0.html#K4">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K4">=&gt;</a> <font color="Maroon">j</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="filter_0.html#K4">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">j</font> )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T9">FILTER_0:9</a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_0.html#K9">&lt;=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a></i>;<br/>
<a NAME="E8:26"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span>,<font color="Maroon">j</font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#D12">FILTER_0:def 12</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">j</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span>,<font color="Maroon">j</font>
<b>by </b><i>, <a class="ref" href="eqrel_1.html#T44">EQREL_1:44</a></i>
<br/>.= 
<font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font>
<b>by </b><i/>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3">LATTICES:def 3</a><br/>


</div>
