<?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">D1</font>, <font color="Olive" title="b2">D2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b3">D1'</font>, <font color="Olive" title="b4">D2'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> holds <br/> ( <font color="Olive" title="b1">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Olive" title="b2">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b2">D2</font> &amp; <font color="Olive" title="b3">D1'</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Olive" title="b4">D2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b3">D1'</font></span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b4">D2'</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b3">D1'</font></span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b4">D2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">D1'</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b4">D2'</font> )</span><br/><b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> for <font color="Olive" title="b1">D1'</font>, <font color="Olive" title="b2">D2'</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> holds <br/> ( <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Olive" title="b1">D1'</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Olive" title="b2">D2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b1">D1'</font></span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b2">D2'</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b1">D1'</font></span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Olive" title="b2">D2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">D1'</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b2">D2'</font> )</span><br/><b>let </b><font color="Maroon" title="c4">D1'</font>, <font color="Maroon" title="c5">D2'</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c4">D1'</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c5">D2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</font></span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</font></span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1'</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c5">D2'</font> )</span><br/>









<a NAME="E1:69"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">L</font> being  <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a><br/> for <font color="Olive" title="b2">D1</font>, <font color="Olive" title="b3">D2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b1">L</font> holds  <font color="Olive" title="b2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Olive" title="b3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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">D1</font>, <font color="Olive" title="b2">D2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">L</font> holds  <font color="Olive" title="b1">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Olive" title="b2">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><br/><b>let </b><font color="Maroon" title="c7">D1</font>, <font color="Maroon" title="c8">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c6">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><br/>





<b>thus </b><a NAME="E1:69_1"/>
<font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </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="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:69_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c10">p</font>, <font color="Maroon" title="c11">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c6">L</font><b> such that </b><br/><a NAME="E3:69_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c11">q</font> &amp; <font color="Maroon" title="c10">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font> &amp; <font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D2</font> )
 ;<br/>
<a NAME="E4:69_1_1"/>
( <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  &amp; <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  &amp; <font color="Maroon" title="c10">p</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c11">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">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="c11">q</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> )
 
;<br/>
<b>hence </b><a NAME="E5:69_1_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span>
 <b>by </b><i><a class="txt" href="filter_2.html#E3:69_1_1"><i><font color="Green" title="E28">A2</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>thus </b><a NAME="E2:69_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font>
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font> )</span><br/>

<b>assume </b><a NAME="E1:69_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </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="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font></span><br/>

<b>then consider </b><font color="Maroon" title="c10">p'</font>, <font color="Maroon" title="c11">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="c6">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span><b> such that </b><br/><a NAME="E3:69_1_2"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">q'</font> &amp; <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font> &amp; <font color="Maroon" title="c11">q'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">D2</font> )
 ;<br/>
<a NAME="E4:69_1_2"/>
( <font color="Maroon" title="c10">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c10">p'</font> &amp; <font color="Maroon" title="c11">q'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c11">q'</font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c10">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="c11">q'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p'</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c11">q'</font> )
 
;<br/>
<b>hence </b><a NAME="E5:69_1_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c8">D2</font>
 <b>by </b><i><a class="txt" href="filter_2.html#E3:69_1_2"><i><font color="Green" title="E28">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>


</div><b>end;</b></div>
<a NAME="E2:69"/><i><font color="Green" title="E28">A4</font></i>: 
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">L</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  &amp; <font color="Maroon" title="c2">D1</font> <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">D1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </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">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  &amp; <font color="Maroon" title="c3">D2</font> <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="c3">D2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </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="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  &amp;  <a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1'</font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</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>  <a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</font> &amp;  <a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">D2'</font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</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>  <a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</font> &amp; <font color="Maroon" title="c4">D1'</font> <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="c4">D1'</font> &amp; <font color="Maroon" title="c5">D2'</font> <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="c5">D2'</font> )
 
;<br/>
<a NAME="E3:69"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">D2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">D1'</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">D2'</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</font></span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="filter_2.html#E59"><i><font color="Green" title="E25">Lm2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:69"/>
( <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c3">D2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c4">D1'</font> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <font color="Maroon" title="c5">D2'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</font></span>)</span> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c4">D1'</font></span>)</span> <a href="filter_2.html#K20" title="FILTER_2:func.20">"\/"</a> <span class="p1">(<span class="default"><a href="filter_2.html#K8" title="FILTER_2:func.8">.:</a> <font color="Maroon" title="c5">D2'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">D1'</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c5">D2'</font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E1:69"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="filter_2.html#E2:69"><i><font color="Green" title="E28">A4</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>
