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<b>let </b><font color="Maroon">I</font> be   <a href="filter_0.html#NM2">I_Lattice</a>;<br/><b>let </b><font color="Maroon">FI</font> be    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">I</font>;<br/>



<b>assume </b><a NAME="E1:28"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">I</font> is <a href="lattices.html#V13">lower-bounded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">j</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> holds <br/> ( <font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Olive">j</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> )
 <b>by </b><i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a></i>;<br/>
<a NAME="E4:28"/><i><font color="Green">E76</font></i>: 
 <a href="lattices.html#K5">Bottom</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="lattices.html#D16">LATTICES:def 16</a></i>;<br/>
<b>set </b><font color="Maroon">L</font> = <font color="Maroon">I</font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">FI</font>;<br/>
<b>set </b><font color="Maroon">R</font> =  <a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font>;<br/>
<b>set </b><font color="Maroon">x</font> = <font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font>;<br/>
<div><i><font color="Green">E78</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">y</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span>;<br/><a NAME="E1:28_1"/>
<font color="Maroon">I</font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">FI</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <span class="p1">(<span class="default"><a href="eqrel_1.html#K9">Class</a> <span class="p2">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">I</font> <a href="filter_1.html#K3" target="_self">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span></span>)</span>,<span class="p1">(<span class="default">the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">I</font> <a href="filter_1.html#K3" target="_self">/\/</a> <span class="p2">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span></span>)</span> #)
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">j</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:28_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="filter_0.html#K11">equivalence_wrt</a> <font color="Maroon">FI</font></span>)</span>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T45">EQREL_1:45</a></i>;<br/><a NAME="E4:28_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">i</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font> )
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E5:28_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font>
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:28_1"/>
<font color="Maroon">y</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:28"/>
<font color="Maroon">I</font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">FI</font> is   <a href="lattices.html#NM4">0_Lattice</a>
 <b>by </b><i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a></i>;<br/>

<b>hence </b><a NAME="E7:28"/>
 <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">I</font> <a href="filter_1.html#K4" target="_self">/\/</a> <font color="Maroon">FI</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">I</font></span>)</span> <a href="filter_1.html#K5" target="_self">/\/</a> <font color="Maroon">FI</font>
 <b>by </b><i>, , <a class="ref" href="lattices.html#D16">LATTICES:def 16</a></i>;<br/>


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