<?xml version="1.0"?>
<div class="add">

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">I</font>, <font color="Olive" title="b2">J</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L</font> holds  <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b3">r</font> where <font color="Olive" title="b3">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b4">p</font>, <font color="Olive" title="b5">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b3">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">I</font> &amp; <font color="Olive" title="b5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">J</font> ) </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c2">I</font>, <font color="Maroon" title="c3">J</font> be    <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span> </span><br/>





<a NAME="E1:73"/>
( <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c3">J</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font> )
 
<b>by </b><i><a class="ref" href="filter_2.html#T37" target="_self" title="FILTER_2:th.37">Th37</a></i>;<br/>
<a NAME="E2:73"/><b>then </b><i><font color="Green" title="E27">A1</font></i>: 
<span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r'</font> where <font color="Olive" title="b1">r'</font> is   <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">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> :  ex <font color="Olive" title="b2">p'</font>, <font color="Olive" title="b3">q'</font> being  <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">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> st <br/>( <font color="Olive" title="b2">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b3">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">r'</font> &amp; <font color="Olive" title="b2">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_0.html#T48" title="FILTER_0:th.48">FILTER_0:48</a></i>;<br/>
<b>thus </b><a NAME="E3:73"/>
<span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <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="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:73_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span> </span><br/>

<b>then consider </b><font color="Maroon" title="c5">r'</font> being   <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">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> &amp;  ex <font color="Olive" title="b1">p'</font>, <font color="Olive" title="b2">q'</font> being  <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">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> st <br/>( <font color="Olive" title="b1">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Olive" title="b2">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">r'</font> &amp; <font color="Olive" title="b1">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) )
 <b>by </b><i><a class="txt" href="filter_2.html#E2:73"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">p'</font>, <font color="Maroon" title="c7">q'</font> being   <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">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E5:73_1"/><i><font color="Green" title="E29">A3</font></i>: 
( <font color="Maroon" title="c6">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">r'</font> &amp; <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E3:73_1"><i><font color="Green" title="E28">A2</font></i></a></i>;<br/>
<a NAME="E6:73_1"/>
(  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c5">r'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a>  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q'</font></span>)</span> &amp;  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q'</font> &amp; <font color="Maroon" title="c6">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c6">p'</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c7">q'</font></span>)</span> &amp;  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p'</font> &amp;  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c7">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q'</font> &amp;  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c5">r'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r'</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E5:73_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="lattice2.html#T54" title="LATTICE2:th.54">LATTICE2:54</a></i>;<br/>
<b>hence </b><a NAME="E7:73_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="filter_2.html#E3:73_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="filter_2.html#E5:73_1"><i><font color="Green" title="E29">A3</font></i></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>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span>  or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )</span><br/>

<b>assume </b><a NAME="E4:73"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">r</font> where <font color="Olive" title="b1">r</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> :  ex <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">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="E6:73"/><i><font color="Green" title="E28">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> &amp;  ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Maroon" title="c5">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">q</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">q</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="E8:73"/><i><font color="Green" title="E29">A5</font></i>: 
( <font color="Maroon" title="c5">r</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> &amp; <font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E6:73"><i><font color="Green" title="E28">A4</font></i></a></i>;<br/>
<a NAME="E9:73"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c5">r</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c6">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> &amp; <font color="Maroon" title="c6">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c7">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> &amp; <font color="Maroon" title="c5">r</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E8:73"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="lattice2.html#T53" title="LATTICE2:th.53">LATTICE2:53</a></i>;<br/>
<b>hence </b><a NAME="E10:73"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">J</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span>
 <b>by </b><i><a class="txt" href="filter_2.html#E2:73"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="filter_2.html#E6:73"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="filter_2.html#E8:73"><i><font color="Green" title="E29">A5</font></i></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>
