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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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> holds <br/> ( <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> iff <font color="Olive" title="b1">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">q</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> iff <font color="Maroon" title="c2">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  )</span><br/>





<a NAME="E1:16"/>
( ( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> implies <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font> ) &amp; ( <span class="p1"><a href="filter_1.html#K9" title="FILTER_1:func.9">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="filter_1.html#K9" title="FILTER_1:func.9">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> ) &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  &amp;  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c1">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font>),<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> #) )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/>
<b>hence </b><a NAME="E2:16"/>
( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> iff <font color="Maroon" title="c2">p</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c3">q</font> <a href="lattice3.html#K4" title="LATTICE3:func.4">%</a>  )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</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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