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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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> holds   <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</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"><font color="Olive" title="b1">p</font></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> <font color="Olive" title="b1">p</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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_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"><font color="Maroon" title="c2">p</font></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> <font color="Maroon" title="c2">p</font></span><br/>



<span class="kw">consider </span><font color="Maroon" title="c3">q9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:72"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
 for <font color="Olive" title="b1">r9</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span> holds <br/> ( <font color="Maroon" title="c3">q9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">r9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q9</font> &amp; <font color="Olive" title="b1">r9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q9</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D13" onmouseover="rs('lattices/D13')" onmouseout="rh()">LATTICES:def 13</a></span>;<br/>
<a NAME="E3:72"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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"><font color="Maroon" title="c2">p</font></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> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T49" target="_self" onmouseover="rs('filter_0/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">p9</font> = <font color="Maroon" title="c2">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T16" target="_self" onmouseover="rs('filter_0/T16')" onmouseout="rh()">Th16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c3">q9</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T49" target="_self" onmouseover="rs('filter_0/T49')" onmouseout="rh()">Th49</a></span>;<br/>
<a NAME="E6:72"/>
<font color="Maroon" title="c3">q9</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E48">A1</font></span></a></span>;<br/>
<a NAME="E7:72"/><span class="kw">then </span>
<font color="Maroon" title="c3">q9</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T4" onmouseover="rs('lattices/T4')" onmouseout="rh()">LATTICES:4</a></span>;<br/>
<a NAME="E8:72"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A2</font></span>: 
<font color="Maroon" title="c5">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_0.html#T51" target="_self" onmouseover="rs('filter_0/T51')" onmouseout="rh()">Th51</a></span>;<br/>
<a NAME="E9:72"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T15" target="_self" onmouseover="rs('filter_0/T15')" onmouseout="rh()">Th15</a></span>;<br/>
<a NAME="E10:72"/>
<font color="Maroon" title="c3">q9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</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"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:72"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="ref" href="rlsub_2.html#T64" onmouseover="rs('rlsub_2/T64')" onmouseout="rh()">RLSUB_2:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:72"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</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"><font color="Maroon" title="c2">p</font></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> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:72"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E9:72"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="ref" href="lattices.html#T8" onmouseover="rs('lattices/T8')" onmouseout="rh()">LATTICES:8</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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