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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">FB</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.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#NM1" title="FILTER_0:NM.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/><span class="kw">let </span><font color="Maroon" title="c2">FB</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="FILTER_0:NM.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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#NM1" title="FILTER_0:NM.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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  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="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/><span class="kw">let </span><font color="Maroon" title="c7">FI</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c3">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



















<span class="kw">thus </span><a NAME="E1:94"/>
( <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="94_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:94_1"/>
( <font color="Maroon" title="c4">i</font> <a href="filter_0.html#K7" title="FILTER_0:func.7">&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#K7" title="FILTER_0:func.7">&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> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D12" target="_self" title="FILTER_0:def.12">FILTER_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">hence </span><a NAME="E2:94_1"/>
<font color="Maroon" title="c4">i</font> <a href="filter_0.html#K7" title="FILTER_0:func.7">&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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T62" target="_self" title="FILTER_0:th.62">Th78</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D12" target="_self" title="FILTER_0:def.12">FILTER_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:94"/>
( <font color="Maroon" title="c8">a</font> <a href="filter_0.html#K7" title="FILTER_0:func.7">&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#K7" title="FILTER_0:func.7">&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> )
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D12" target="_self" title="FILTER_0:def.12">FILTER_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">hence </span><a NAME="E3:94"/>
<font color="Maroon" title="c8">a</font> <a href="filter_0.html#K7" title="FILTER_0:func.7">&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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T62" target="_self" title="FILTER_0:th.62">Th78</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_0.html#D12" target="_self" title="FILTER_0:def.12">FILTER_0:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
