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

<b>let </b><font color="Maroon" title="c1">I</font> be   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FI</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">I</font><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b2">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Olive" title="b1">FI</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Olive" title="b1">FI</font> iff <font color="Olive" title="b2">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Olive" title="b3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">FI</font> )</span><br/><b>let </b><font color="Maroon" title="c2">FI</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Olive" title="b1">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> iff <font color="Olive" title="b1">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Olive" title="b2">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> )</span><br/><b>let </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> iff <font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> )</span><br/>







<b>set </b><font color="Maroon" title="c5">R</font> =  <a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font>;<br/>
<b>set </b><font color="Maroon" title="c6">a</font> = <font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font>;<br/>
<b>set </b><font color="Maroon" title="c7">b</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font>;<br/>
<b>thus </b><a NAME="E1:26"/>
( <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> implies <font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> implies <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> )</span><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" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font></span><br/>

<a NAME="E2:26_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> &amp;  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> &amp; <font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span>,<font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> &amp; <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span>,<font color="Maroon" title="c4">j</font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a>, <a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a>, <a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></i>;<br/>
<a NAME="E3:26_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span>,<font color="Maroon" title="c4">j</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="eqrel_1.html#T30" title="EQREL_1:th.30">EQREL_1:30</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#T22" title="LATTICES:th.22">LATTICES:22</a>, <a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></i>;<br/>
<a NAME="E4:26_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D12" title="FILTER_0:def.12">FILTER_0:def 12</a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<a NAME="E5:26_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D1" title="FILTER_0:def.1">FILTER_0:def 1</a>, <a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E6:26_1"/>
<font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T9" title="FILTER_0:th.9">FILTER_0:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E2:26"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span><br/>

<a NAME="E3:26"/>
( <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> &amp; <font color="Maroon" title="c4">j</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T2" title="FILTER_0:th.2">FILTER_0:2</a>, <a class="ref" href="filter_0.html#T3" title="FILTER_0:th.3">FILTER_0:3</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#T17" title="LATTICES:th.17">LATTICES:17</a>, <a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></i>;<br/>
<a NAME="E4:26"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c4">j</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font> &amp; <font color="Maroon" title="c1">I</font> is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T4" title="FILTER_0:th.4">FILTER_0:4</a>, <a class="ref" href="filter_0.html#T37" title="FILTER_0:th.37">FILTER_0:37</a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></i>;<br/>
<a NAME="E5:26"/><b>then </b>
( <font color="Maroon" title="c3">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">I</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">j</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> &amp;  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">I</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T12" title="FILTER_0:th.12">FILTER_0:12</a>, <a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></i>;<br/>
<a NAME="E6:26"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> &amp; <font color="Maroon" title="c4">j</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">j</font> )
 
<b>by </b><i><a class="txt" href="#E2:26"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="filter_0.html#T9" title="FILTER_0:th.9">FILTER_0:9</a></i>;<br/>
<a NAME="E7:26"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FI</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#D1" title="FILTER_0:def.1">FILTER_0:def 1</a></i>;<br/>
<a NAME="E8:26"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span>,<font color="Maroon" title="c4">j</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#D12" title="FILTER_0:def.12">FILTER_0:def 12</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font>
<b>by </b><i><a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></i>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">j</font></span>)</span>
<b>by </b><i><a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></i>
<br/>.= 
 <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span>,<font color="Maroon" title="c4">j</font>
<b>by </b><i><a class="txt" href="#E8:26"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></i>
<br/>.= 
<font color="Maroon" title="c4">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font>
<b>by </b><i><a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
