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<b>let </b><font color="Maroon" title="c1">I</font> be   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">FI</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">I</font>  st <font color="Maroon" title="c1">I</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> holds <br/>( <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Olive" title="b1">FI</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Olive" title="b1">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Olive" title="b1">FI</font> )</span><br/><b>let </b><font color="Maroon" title="c2">FI</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">I</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">I</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> implies ( <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> ) )</span><br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c1">I</font> is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a> &amp;  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> )</span><br/>

<b>then consider </b><font color="Maroon" title="c3">i</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green" title="E22">A2</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font> holds <br/> ( <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Olive" title="b1">j</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> )
 <b>by </b><i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green" title="E23">A3</font></i>: 
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="txt" href="#E1:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="#E3:28"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="lattices.html#D16" title="LATTICES:def.16">LATTICES:def 16</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">L</font> = <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font>;<br/>
<b>set </b><font color="Maroon" title="c5">R</font> =  <a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font>;<br/>
<b>set </b><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font>;<br/>
<div><i><font color="Green" title="E24">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> &amp; <font color="Maroon" title="c7">y</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> )</span><br/><a NAME="E1:28_1"/>
<font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K8" title="EQREL_1:func.8">Class</a> <span class="p2">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">I</font> <a href="filter_1.html#K3" title="FILTER_1:func.3">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span></span>)</span> #)
 <b>by </b><i><a class="ref" href="filter_1.html#D5" target="_self" title="FILTER_1:def.5">Def5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">I</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="eqrel_1.html#K6" title="EQREL_1:func.6">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11" title="FILTER_0:func.11">equivalence_wrt</a> <font color="Maroon" title="c2">FI</font></span>)</span>,<font color="Maroon" title="c8">j</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45" title="EQREL_1:th.45">EQREL_1:45</a></i>;<br/><a NAME="E4:28_1"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font> )
 <b>by </b><i><a class="txt" href="#E3:28"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="#E3:28_1"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="filter_1.html#D6" target="_self" title="FILTER_1:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E5:28_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font>
 <b>by </b><i><a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span><br/><b>thus </b><a NAME="E6:28_1"/>
<font color="Maroon" title="c7">y</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i</font> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font>
 <b>by </b><i><a class="txt" href="#E4:28_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="filter_1.html#T15" target="_self" title="FILTER_1:th.15">Th15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28"/>
<font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font> is   <a href="lattices.html#NM4" title="LATTICES:NM.4">0_Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D13" title="LATTICES:def.13">LATTICES:def 13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font></span><br/>

<b>hence </b><a NAME="E7:28"/>
 <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I</font> <a href="filter_1.html#K4" title="FILTER_1:func.4">/\/</a> <font color="Maroon" title="c2">FI</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">I</font></span>)</span> <a href="filter_1.html#K5" title="FILTER_1:func.5">/\/</a> <font color="Maroon" title="c2">FI</font>
 <b>by </b><i><a class="txt" href="#E4:28"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="#E5:28"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="lattices.html#D16" title="LATTICES:def.16">LATTICES:def 16</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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