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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">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">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">F</font> being  <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b3">F</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</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#K10" title="FILTER_0:func.10">equivalence_wrt</a> <font color="Olive" title="b3">F</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</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">F</font> being  <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Olive" title="b1">F</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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#K10" title="FILTER_0:func.10">equivalence_wrt</a> <font color="Olive" title="b1">F</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">F</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</a> of <font color="Maroon" title="c1">L</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="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c4">F</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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#K10" title="FILTER_0:func.10">equivalence_wrt</a> <font color="Maroon" title="c4">F</font> )</span><br/>







<a NAME="E1:106"/>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c4">F</font> iff <font color="Maroon" title="c2">p</font> <a href="filter_0.html#K9" title="FILTER_0:func.9">&lt;=&gt;</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D13" target="_self" title="FILTER_0:def.13">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:106"/>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="filter_0.html#R1" title="FILTER_0:pred.1">are_equivalence_wrt</a> <font color="Maroon" title="c4">F</font> iff <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</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#K10" title="FILTER_0:func.10">equivalence_wrt</a> <font color="Maroon" title="c4">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#D12" target="_self" title="FILTER_0:def.12">Def12</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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