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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="firebrick">::  thesis: </font></span></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#V13" title="LATTICES:attr.13">lower-bounded</a>  holds <br/><span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Olive" title="b1">p</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span></span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>  implies <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span> )</span><br/>



<span class="kw">assume </span><a NAME="E1:88"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c1">L</font> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span></span><br/>

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</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#K9" title="FILTER_2:func.9">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span> )</span><br/>

<a NAME="E2:88"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
( <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#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> iff <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T28" target="_self" title="FILTER_2:th.28">Th29</a></span>;<br/>
<a NAME="E3:88"/>
(  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:88"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="lattices.html#T16" title="LATTICES:th.16">LATTICES:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:88"/>
( <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#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</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#K9" title="FILTER_2:func.9">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span>)</span>,<font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">#]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:88"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="filter_2.html#T62" target="_self" title="FILTER_2:th.62">Th63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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