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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"> ( <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#K9" title="FILTER_2:func.9">(.</a><span class="default"><font color="Maroon" title="c1">L</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><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#K10" title="FILTER_2:func.10">.&gt;</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:49"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">(.</a><span class="default"><font color="Maroon" title="c1">L</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><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#K10" title="FILTER_2:func.10">.&gt;</a></span></span><br/>

<a NAME="E2:49"/><span class="kw">then </span>
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T64" title="LATTICE2:th.64">LATTICE2:64</a></span>;<br/>
<a NAME="E3:49"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A2</font></span>: 
<span class="p1"><a href="filter_2.html#K1" title="FILTER_2:func.1">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span></span><a href="filter_2.html#K1" title="FILTER_2:func.1">.)</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span></span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T20" title="FILTER_0:th.20">FILTER_0:20</a></span>;<br/>
<a NAME="E4:49"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:49"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="lattice2.html#T79" title="LATTICE2:th.79">LATTICE2:79</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:49"/>
<span class="p1"><a href="filter_2.html#K9" title="FILTER_2:func.9">(.</a><span class="default"><font color="Maroon" title="c1">L</font></span><a href="filter_2.html#K9" title="FILTER_2:func.9">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><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#K10" title="FILTER_2:func.10">.&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:49"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="filter_2.html#T30" target="_self" title="FILTER_2:th.30">Th30</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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