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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">F</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:74"/><i><font color="Green" title="E52">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">p'</font>, <font color="Maroon" title="c4">q'</font> be   <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#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7" target="_self" title="FILTER_0:def.7">FILTER_0:def 7</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">p</font> = <font color="Maroon" title="c3">p'</font>, <font color="Maroon" title="c6">q</font> = <font color="Maroon" title="c4">q'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <b>by </b><i><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E4:74"/><i><font color="Green" title="E53">A2</font></i>: 
( <font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font> &amp; ( for <font color="Olive" title="b1">r1</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="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b1">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font> holds <br/><font color="Olive" title="b1">r1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">r</font> ) )
 <b>by </b><i><a class="txt" href="filter_0.html#E1:74"><i><font color="Green" title="E52">A1</font></i></a>, <a class="ref" href="filter_0.html#D7" target="_self" title="FILTER_0:def.7">Def7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">I</font> = <font color="Maroon" title="c1">L</font> as   <a href="filter_0.html#NM2" title="FILTER_0:NM.2">I_Lattice</a> <b>by </b><i><a class="txt" href="filter_0.html#E1:74"><i><font color="Green" title="E52">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">i</font> = <font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c6">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c8">I</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">FI</font> = <font color="Maroon" title="c2">F</font> as    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c8">I</font> ;<br/>
<a NAME="E8:74"/>
( <font color="Maroon" title="c9">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c9">i</font> <a href="filter_0.html#K4" title="FILTER_0:func.4">=&gt;</a> <font color="Maroon" title="c10">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">FI</font> )
 
<b>by </b><i><a class="txt" href="filter_0.html#E4:74"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="filter_0.html#D8" target="_self" title="FILTER_0:def.8">Def8</a>, <a class="ref" href="filter_0.html#T42" target="_self" title="FILTER_0:th.42">Th42</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">r'</font> = <font color="Maroon" title="c7">r</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#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font></span>)</span> <b>by </b><i><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c12">r'</font>
;<br/>

<a NAME="E10:74"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">r'</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></i>;<br/>
<b>hence </b><a NAME="E11:74"/>
<font color="Maroon" title="c3">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c12">r'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q'</font>
 <b>by </b><i><a class="txt" href="filter_0.html#E4:74"><i><font color="Green" title="E53">A2</font></i></a>, <a class="ref" href="filter_0.html#T65" target="_self" title="FILTER_0:th.65">Th65</a></i>;<br/>

<b>let </b><font color="Maroon" title="c13">s'</font> be   <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#K8" title="FILTER_0:func.8">latt</a> <font color="Maroon" title="c2">F</font></span>)</span>;<br/>

<b>assume </b><a NAME="E12:74"/><i><font color="Green" title="E54">A3</font></i>: 
<font color="Maroon" title="c3">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">q'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c14">s</font> = <font color="Maroon" title="c13">s'</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">F</font> <b>by </b><i><a class="ref" href="filter_0.html#T63" target="_self" title="FILTER_0:th.63">Th63</a></i>;<br/>
<a NAME="E14:74"/>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c13">s'</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T64" target="_self" title="FILTER_0:th.64">Th64</a></i>;<br/>
<a NAME="E15:74"/><b>then </b>
<font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c14">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">q</font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E12:74"><i><font color="Green" title="E54">A3</font></i></a>, <a class="ref" href="filter_0.html#T65" target="_self" title="FILTER_0:th.65">Th65</a></i>;<br/>
<a NAME="E16:74"/><b>then </b>
<font color="Maroon" title="c14">s</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">r</font>
 
<b>by </b><i><a class="txt" href="filter_0.html#E4:74"><i><font color="Green" title="E53">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:74"/>
<font color="Maroon" title="c13">s'</font> <a href="lattices.html#R1" title="LATTICES:pred.1">[=</a> <font color="Maroon" title="c12">r'</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T65" target="_self" title="FILTER_0:th.65">Th65</a></i>;<br/>


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