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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> 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/>(  <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>  &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )</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="Red">::  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 (  <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>  &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:112"/><span class="lab"><font color="Green" title="E45">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="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>  &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )</span><br/>

<a NAME="E2:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A2</font></span>: 
<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice2.html#T63" title="LATTICE2:th.63">LATTICE2:63</a></span>;<br/>
<a NAME="E3:112"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A3</font></span>: 
 <a href="filter_0.html#K8" title="FILTER_0:func.8">latt</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"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> is  <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T66" title="FILTER_0:th.66">FILTER_0:66</a></span>;<br/>
<span class="lab"><font color="Green" title="E48">A4</font></span>:  <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> = 
<span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <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 class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span>)</span> <a href="filter_2.html#K16" title="FILTER_2:func.16">.:</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T71" target="_self" title="FILTER_2:th.71">Th71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <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 class="p2"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span></span>)</span> <a href="filter_2.html#K16" title="FILTER_2:func.16">.:</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T30" target="_self" title="FILTER_2:th.30">Th30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p2"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span></span>)</span> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T70" target="_self" title="FILTER_2:th.70">Th70</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:112"/>
 <a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> is  <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="lattice2.html#T64" title="LATTICE2:th.64">LATTICE2:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E6:112"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_0.html#K8" title="FILTER_0:func.8">latt</a> <span class="p2"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K6" title="LATTICES:func.6">Top</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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:112"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="filter_0.html#T72" title="FILTER_0:th.72">FILTER_0:72</a></span>;<br/>
<span class="kw">hence </span> <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><a href="filter_2.html#K15" title="FILTER_2:func.15">latt</a> <font color="Maroon" title="c1">L</font>,<span class="p2"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span></span>)</span> = 
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:112"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="txt" href="#E3:112"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="ref" href="lattice2.html#T79" title="LATTICE2:th.79">LATTICE2:79</a></span>
<br/>.= 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:112"><span class="lab"><font color="Green" title="E45">A1</font></span></a>, <a class="ref" href="lattice2.html#T78" title="LATTICE2:th.78">LATTICE2:78</a></span>
;<br/> <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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