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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">FI</font> being  <a href="filter_0.html#NM1" title="FILTER_0:NM.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/><span class="kw">let </span><font color="Maroon" title="c2">FI</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>







<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<a NAME="E1:26"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T2" title="FILTER_0:th.2">FILTER_0:2</a></span>;<br/>
<a NAME="E2:26"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T17" title="LATTICES:th.17">LATTICES:17</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:26"/>
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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>
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></span>;<br/>
<a NAME="E3:26_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></span>;<br/>
<a NAME="E4:26_1"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></span>;<br/>
<a NAME="E5:26_1"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></span>;<br/>
<a NAME="E6:26_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="eqrel_1.html#T28" title="EQREL_1:th.28">EQREL_1:28</a></span>;<br/>
<a NAME="E7:26_1"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></span>;<br/>
<a NAME="E8:26_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E5:26_1"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E3:26_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E6:26_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="eqrel_1.html#T30" title="EQREL_1:th.30">EQREL_1:30</a></span>;<br/>
<a NAME="E9:26_1"/><span class="kw">then </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_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D12" title="FILTER_0:def.12">FILTER_0:def 12</a></span>;<br/>
<a NAME="E10:26_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A8</font></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_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 class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T8" title="FILTER_0:th.8">FILTER_0:8</a></span>;<br/>
<a NAME="E11:26_1"/><span class="lab"><font color="Green" title="E27">A9</font></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="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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></span>;<br/>
<a NAME="E12:26_1"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></span>;<br/>
<a NAME="E13:26_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E4:26_1"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="lattices.html#T25" title="LATTICES:th.25">LATTICES:25</a></span>;<br/>
<a NAME="E14:26_1"/><span class="kw">then </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_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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:26_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="filter_0.html#T9" title="FILTER_0:th.9">FILTER_0:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E4:26"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T3" title="FILTER_0:th.3">FILTER_0:3</a></span>;<br/>
<a NAME="E5:26"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T2" title="FILTER_0:th.2">FILTER_0:2</a></span>;<br/>
<a NAME="E6:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A10</font></span>: 
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></span>;<br/>
<a NAME="E7:26"/>
<font color="Maroon" title="c1">I</font> is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T37" title="FILTER_0:th.37">FILTER_0:37</a></span>;<br/>
<a NAME="E8:26"/><span class="kw">then </span>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T12" title="FILTER_0:th.12">FILTER_0:12</a></span>;<br/>
<a NAME="E9:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A11</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E21">A10</font></span></a>, <a class="ref" href="filter_0.html#T9" title="FILTER_0:th.9">FILTER_0:9</a></span>;<br/>
<a NAME="E10:26"/><span class="lab"><font color="Green" title="E23">A12</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></span>;<br/>
<a NAME="E11:26"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></span>;<br/>
<a NAME="E12:26"/><span class="kw">then </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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E23">A12</font></span></a>, <a class="ref" href="filter_0.html#T4" title="FILTER_0:th.4">FILTER_0:4</a></span>;<br/>
<a NAME="E13:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A13</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E14:26"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E15:26"/><span class="kw">then </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_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 class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26"><span class="lab"><font color="Green" title="E24">A13</font></span></a>, <a class="ref" href="filter_0.html#T9" title="FILTER_0:th.9">FILTER_0:9</a></span>;<br/>
<a NAME="E16:26"/><span class="kw">then </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_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E22">A11</font></span></a>, <a class="ref" href="filter_0.html#T8" title="FILTER_0:th.8">FILTER_0:8</a></span>;<br/>
<a NAME="E17:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A14</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D12" title="FILTER_0:def.12">FILTER_0:def 12</a></span>;<br/>
<span class="kw">thus </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:26"><span class="lab"><font color="Green" title="E25">A14</font></span></a>, <a class="ref" href="eqrel_1.html#T44" title="EQREL_1:th.44">EQREL_1:44</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></span>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
