<?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> holds  <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Olive" title="b1">FI</font> is <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a></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"> <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font> is <a href="filter_0.html#V3" title="FILTER_0:attr.3">implicative</a></span><br/>



<b>set </b><font color="Maroon" title="c3">L</font> = <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font>;<br/>
<b>set </b><font color="Maroon" title="c4">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>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7" title="FILTER_0:def.7">FILTER_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> st <br/>( <font color="Maroon" title="c5">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">y</font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c5">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">y</font> or <font color="Olive">b<sub>2</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Olive">b<sub>1</sub></font> ) ) )</span><br/>

<a NAME="E1:30"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<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>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <span class="p2">(<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>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <span class="p2">(<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>)</span> #)
 
<b>by </b><i><a class="ref" href="filter_1.html#D5" target="_self" title="FILTER_1:def.5">Def5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:30"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c5">x</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="c7">i</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E5:30"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c6">y</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="c8">j</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<b>take </b><font color="Maroon" title="c9">z</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</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="c5">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c9">z</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">y</font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c5">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">z</font> ) ) )</span><br/>

<a NAME="E6:30"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c1">I</font> is   <a href="lattices.html#NM5" title="LATTICES:NM.5">1_Lattice</a> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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="txt" href="filter_1.html#E3:30"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="filter_1.html#E5:30"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a>, <a class="ref" href="filter_0.html#T37" title="FILTER_0:th.37">FILTER_0:37</a></i>;<br/>
<a NAME="E7:30"/><b>then </b>
( <font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c8">j</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="filter_0.html#D8" title="FILTER_0:def.8">FILTER_0:def 8</a>, <a class="ref" href="lattices.html#T43" title="LATTICES:th.43">LATTICES:43</a></i>;<br/>
<a NAME="E8:30"/><b>then </b>
(  <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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font> &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="txt" href="filter_1.html#E6:30"><i><font color="Green" title="E26">A4</font></i></a>, <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="E9:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">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>;<br/>
<a NAME="E10:30"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span></span>)</span> <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="c6">y</font>
 
<b>by </b><i><a class="txt" href="filter_1.html#E6:30"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="filter_1.html#T16" target="_self" title="FILTER_1:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E11:30"/>
<font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c9">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E6:30"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span>  holds <br/>(  not <font color="Maroon" title="c5">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">y</font> or <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">z</font> )</span><br/>

<b>let </b><font color="Maroon" title="c10">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c10">t</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c6">y</font> or <font color="Maroon" title="c10">t</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">z</font> )</span><br/>

<b>consider </b><font color="Maroon" title="c11">k</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E13:30"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c10">t</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="c11">k</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E1:30"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/>
<a NAME="E14:30"/><i><font color="Green" title="E28">A6</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">k</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> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">k</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="c11">k</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> <font color="Maroon" title="c10">t</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E13:30"><i><font color="Green" title="E27">A5</font></i></a>, <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></i>;<br/>
<b>assume </b><a NAME="E15:30"/>
<font color="Maroon" title="c5">x</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c10">t</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c10">t</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">z</font></span><br/>

<a NAME="E16:30"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">k</font></span>)</span> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">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="c7">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">k</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">i</font> )
 
<b>by </b><i><a class="txt" href="filter_1.html#E6:30"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="filter_1.html#E14:30"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="filter_1.html#T16" target="_self" title="FILTER_1:th.16">Th16</a></i>;<br/>
<a NAME="E17:30"/><b>then </b>
<font color="Maroon" title="c11">k</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c8">j</font></span>)</span> <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_1.html#T17" target="_self" title="FILTER_1:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E18:30"/>
<font color="Maroon" title="c10">t</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c9">z</font>
 <b>by </b><i><a class="txt" href="filter_1.html#E14:30"><i><font color="Green" title="E28">A6</font></i></a>, <a class="ref" href="filter_1.html#T16" target="_self" title="FILTER_1:th.16">Th16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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