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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/>(  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<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>) 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#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><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>)</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="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 (  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<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>) 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#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><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>)</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="E1:108"/><span class="lab"><font color="Green" title="E44">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"> (  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<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>) 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#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><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>)</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="E2:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">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#T48" title="LATTICE2:th.48">LATTICE2:48</a></span>;<br/>
<a NAME="E3:108"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
 <a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0: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#T52" title="FILTER_0:th.52">FILTER_0:52</a></span>;<br/>
<span class="lab"><font color="Green" title="E47">A4</font></span>:  <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<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>) = 
<span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">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#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#K3" title="FILTER_2:func.3">.:</a></span>)</span>)</span>)</span> <a href="filter_2.html#K11" title="FILTER_2:func.11">.:</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T70" target="_self" title="FILTER_2:th.70">Th71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="filter_2.html#K10" title="FILTER_2:func.10">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_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>)</span>)</span> <a href="filter_2.html#K11" title="FILTER_2:func.11">.:</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T29" target="_self" title="FILTER_2:th.29">Th30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <span class="p2"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0: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#T69" target="_self" title="FILTER_2:th.69">Th70</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:108"/>
 <a href="filter_2.html#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<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>) 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="#E3:108"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="lattice2.html#T49" title="LATTICE2:th.49">LATTICE2:49</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><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>)</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="E6:108"/>
 <a href="lattices.html#K6" title="LATTICES:func.6">Top</a> <span class="p1">(<span class="default"><a href="filter_0.html#K6" title="FILTER_0:func.6">latt</a> <span class="p2"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0: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="#E2:108"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="filter_0.html#T57" title="FILTER_0:th.57">FILTER_0:57</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#K10" title="FILTER_2:func.10">latt</a> (<font color="Maroon" title="c1">L</font>,<span class="p2"><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>)</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="#E4:108"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="txt" href="#E3:108"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="lattice2.html#T62" title="LATTICE2:th.62">LATTICE2:62</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="#E1:108"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="lattice2.html#T61" title="LATTICE2:th.61">LATTICE2:61</a></span>
;<br/> <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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