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<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">k</font>, <font color="Olive" title="b4">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font>  st <font color="Olive" title="b2">i</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="Olive" title="b1">FI</font></span>)</span>,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b4">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="Olive" title="b1">FI</font></span>)</span>,<font color="Olive" title="b3">k</font> holds <br/>( <font color="Olive" title="b2">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b4">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="Olive" title="b1">FI</font></span>)</span>,<font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b2">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b4">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="Olive" title="b1">FI</font></span>)</span>,<font color="Olive" title="b3">k</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">k</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>  st <font color="Olive" title="b1">i</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="Olive" title="b2">k</font> &amp; <font color="Olive" title="b3">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="Olive" title="b2">k</font> holds <br/>( <font color="Olive" title="b1">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">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="Olive" title="b2">k</font> &amp; <font color="Olive" title="b1">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">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="Olive" title="b2">k</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">k</font>, <font color="Maroon" title="c5">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="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">k</font> &amp; <font color="Maroon" title="c5">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">k</font> implies ( <font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">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">k</font> &amp; <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">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">k</font> ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:82"/><span class="lab"><font color="Green" title="E56">A1</font></span>: 
<font color="Maroon" title="c3">i</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">k</font>
 <span class="kw">and </span><br/><a NAME="E2:82"/><span class="lab"><font color="Green" title="E57">A2</font></span>: 
<font color="Maroon" title="c5">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">k</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="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">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">k</font> &amp; <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">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">k</font> )</span><br/>

<a NAME="E3:82"/><span class="lab"><font color="Green" title="E58">A3</font></span>: 
<font color="Maroon" title="c3">i</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">k</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="#E1:82"><span class="lab"><font color="Green" title="E56">A1</font></span></a>, <a class="txt" href="filter_1.html#E72"><span class="lab"><font color="Green" title="E54">Lm4</font></span></a></span>;<br/>
<a NAME="E4:82"/><span class="lab"><font color="Green" title="E59">A4</font></span>: 
<font color="Maroon" title="c5">j</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">k</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="#E2:82"><span class="lab"><font color="Green" title="E57">A2</font></span></a>, <a class="txt" href="filter_1.html#E72"><span class="lab"><font color="Green" title="E54">Lm4</font></span></a></span>;<br/>
<a NAME="E5:82"/>
<font color="Maroon" title="c4">k</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T18" title="LATTICES:th.18">LATTICES:18</a></span>;<br/>
<a NAME="E6:82"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A5</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> <font color="Maroon" title="c5">j</font></span>)</span> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">k</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="#E3:82"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="filter_1.html#T59" target="_self" title="FILTER_1:th.59">Th59</a></span>;<br/>
<a NAME="E7:82"/>
<font color="Maroon" title="c4">k</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">k</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/>
<a NAME="E8:82"/><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="c5">j</font></span>)</span> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c4">k</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="#E3:82"><span class="lab"><font color="Green" title="E58">A3</font></span></a>, <a class="txt" href="#E4:82"><span class="lab"><font color="Green" title="E59">A4</font></span></a>, <a class="ref" href="filter_1.html#T59" target="_self" title="FILTER_1:th.59">Th59</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:82"/>
( <font color="Maroon" title="c3">i</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c5">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">k</font> &amp; <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c5">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">k</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82"><span class="lab"><font color="Green" title="E60">A5</font></span></a>, <a class="txt" href="filter_1.html#E72"><span class="lab"><font color="Green" title="E54">Lm4</font></span></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/>


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