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<b>let </b><font color="Maroon" title="c1">B</font> be   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_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">FB</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">B</font><br/> for <font color="Olive" title="b2">I</font> being  <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font>, <font color="Olive" title="b5">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b2">I</font><br/> for <font color="Olive" title="b6">FI</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Olive" title="b2">I</font><br/> for <font color="Olive" title="b7">a</font>, <font color="Olive" title="b8">b</font>, <font color="Olive" title="b9">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( ( <font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b6">FI</font> &amp; <font color="Olive" title="b4">j</font>,<font color="Olive" title="b5">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b6">FI</font> implies <font color="Olive" title="b3">i</font>,<font color="Olive" title="b5">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b6">FI</font> ) &amp; ( <font color="Olive" title="b7">a</font>,<font color="Olive" title="b8">b</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b1">FB</font> &amp; <font color="Olive" title="b8">b</font>,<font color="Olive" title="b9">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b1">FB</font> implies <font color="Olive" title="b7">a</font>,<font color="Olive" title="b9">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b1">FB</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">FB</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">B</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> being  <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font>, <font color="Olive" title="b4">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">I</font><br/> for <font color="Olive" title="b5">FI</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Olive" title="b1">I</font><br/> for <font color="Olive" title="b6">a</font>, <font color="Olive" title="b7">b</font>, <font color="Olive" title="b8">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( ( <font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b5">FI</font> &amp; <font color="Olive" title="b3">j</font>,<font color="Olive" title="b4">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b5">FI</font> implies <font color="Olive" title="b2">i</font>,<font color="Olive" title="b4">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b5">FI</font> ) &amp; ( <font color="Olive" title="b6">a</font>,<font color="Olive" title="b7">b</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Olive" title="b7">b</font>,<font color="Olive" title="b8">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> implies <font color="Olive" title="b6">a</font>,<font color="Olive" title="b8">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">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">i</font>, <font color="Olive" title="b2">j</font>, <font color="Olive" title="b3">k</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">I</font><br/> for <font color="Olive" title="b4">FI</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c3">I</font><br/> for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( ( <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b4">FI</font> &amp; <font color="Olive" title="b2">j</font>,<font color="Olive" title="b3">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b4">FI</font> implies <font color="Olive" title="b1">i</font>,<font color="Olive" title="b3">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b4">FI</font> ) &amp; ( <font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Olive" title="b6">b</font>,<font color="Olive" title="b7">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> implies <font color="Olive" title="b5">a</font>,<font color="Olive" title="b7">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> ) )</span><br/><b>let </b><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c6">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">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">FI</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c3">I</font><br/> for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( ( <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b1">FI</font> &amp; <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b1">FI</font> implies <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b1">FI</font> ) &amp; ( <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> implies <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> ) )</span><br/><b>let </b><font color="Maroon" title="c7">FI</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c3">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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( ( <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> &amp; <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> implies <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> implies <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> ) )</span><br/><b>let </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</font>, <font color="Maroon" title="c10">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> &amp; <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> implies <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> ) &amp; ( <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> implies <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> ) )</span><br/>



















<b>thus </b><a NAME="E1:95"/>
( <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> &amp; <font color="Maroon" title="c5">j</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font> implies <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">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="c8">a</font>,<font color="Maroon" title="c9">b</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c10">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> implies <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="95_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:95_1"/>
( <font color="Maroon" title="c4">i</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c5">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">FI</font> &amp; <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="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">FI</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D13" target="_self" title="FILTER_0:def.13">FILTER_0:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">k</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c7">FI</font></span><br/>

<b>hence </b><a NAME="E2:95_1"/>
<font color="Maroon" title="c4">i</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">FI</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T78" target="_self" title="FILTER_0:th.78">Th78</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D13" target="_self" title="FILTER_0:def.13">FILTER_0:def 13</a> <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>

<a NAME="E2:95"/><i><font color="Green" title="E65">A1</font></i>: 
<font color="Maroon" title="c1">B</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>
 
<b>by </b><i><a class="ref" href="filter_0.html#T40" target="_self" title="FILTER_0:th.40">Th40</a></i>;<br/>
<b>assume </b><a NAME="E3:95"/>
( <font color="Maroon" title="c8">a</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> &amp; <font color="Maroon" title="c9">b</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D13" target="_self" title="FILTER_0:def.13">FILTER_0:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">a</font>,<font color="Maroon" title="c10">c</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c2">FB</font></span><br/>

<b>hence </b><a NAME="E4:95"/>
<font color="Maroon" title="c8">a</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">FB</font>
 <b>by </b><i><a class="txt" href="#E2:95"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="filter_0.html#T78" target="_self" title="FILTER_0:th.78">Th78</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D13" target="_self" title="FILTER_0:def.13">FILTER_0:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
