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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> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> holds <br/><span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Olive" title="b1">p</font>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span><br/><b>let </b><font color="Maroon" title="c2">p</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="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> implies <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> )</span><br/>



<b>assume </b><a NAME="E1:91"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span></span><br/>

<b>let </b><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>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D1" target="_self" title="FILTER_2:def.1">FILTER_2:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> iff <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> )</span><br/>

<a NAME="E2:91"/>
( <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="txt" href="#E1:91"><i><font color="Green" title="E35">A1</font></i></a>, <a class="ref" href="lattices.html#T45" title="LATTICES:th.45">LATTICES:45</a></i>;<br/>
<a NAME="E3:91"/><b>then </b>
( ( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> 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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> implies <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> ) &amp; ( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> 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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> implies <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> ) )
 
<b>by </b><i><a class="ref" href="filter_2.html#T63" target="_self" title="FILTER_2:th.63">Th63</a>, <a class="ref" href="filter_0.html#T18" title="FILTER_0:th.18">FILTER_0:18</a></i>;<br/>
<b>hence </b><a NAME="E4:91"/>
( <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> iff <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K22" title="FILTER_2:func.22">[#</a><span class="default"><font color="Maroon" title="c2">p</font>,<span class="p2">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="filter_2.html#K22" title="FILTER_2:func.22">#]</a></span> )
 ; <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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