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<div class="add">

<b>let </b><font color="Maroon" title="c3">it1</font>, <font color="Maroon" title="c4">it2</font> be  <a href="lattices.html#V3" title="LATTICES:attr.3">strict</a> <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:31_1_2"/><i><font color="Green" title="E22">A19</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font>  ex <font color="Olive" title="b3">x'</font>, <font color="Olive" title="b4">y'</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x'</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y'</font> &amp; ( <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">y</font> implies <font color="Olive" title="b3">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">y'</font> ) &amp; ( <font color="Olive" title="b3">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">y'</font> implies <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">y</font> ) ) ) )
 <b>and </b><br/><a NAME="E2:31_1_2"/><i><font color="Green" title="E23">A20</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font>  ex <font color="Olive" title="b3">x'</font>, <font color="Olive" title="b4">y'</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">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">x'</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">y'</font> &amp; ( <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">y</font> implies <font color="Olive" title="b3">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">y'</font> ) &amp; ( <font color="Olive" title="b3">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">y'</font> implies <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">y</font> ) ) ) )
 ;<br/>

<a NAME="E3:31_1_2"/><i><font color="Green" title="E24">A21</font></i>: 
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">it1</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c3">it1</font></span>)</span> #)
 
;<br/>
<a NAME="E4:31_1_2"/><i><font color="Green" title="E25">A22</font></i>: 
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">it2</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c4">it2</font></span>)</span> #)
 
;<br/>
<a NAME="E5:31_1_2"/><i><font color="Green" title="E26">A23</font></i>: 
 <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font> : <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_1.html#D8" title="FILTER_1:def.8">FILTER_1:def 8</a></i>;<br/>
<a NAME="E6:31_1_2"/><i><font color="Green" title="E27">A24</font></i>: 
 <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c4">it2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> where <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font> : <font color="Olive" title="b1">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b2">q</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_1.html#D8" title="FILTER_1:def.8">FILTER_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_2_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c3">it1</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font><b> such that </b><br/><a NAME="E3:31_1_2_1_1"/><i><font color="Green" title="E28">A25</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">p1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font> )
 <b>by </b><i><a class="txt" href="knaster.html#E5:31_1_2"><i><font color="Green" title="E26">A23</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">p1</font> = <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c9">q1</font> = <font color="Maroon" title="c7">q1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font> ;<br/><b>consider </b><font color="Maroon" title="c10">pl1</font>, <font color="Maroon" title="c11">ql1</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="E6:31_1_2_1_1"/><i><font color="Green" title="E29">A26</font></i>: 
( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pl1</font> &amp; <font color="Maroon" title="c9">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ql1</font> &amp; ( <font color="Maroon" title="c8">p1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">q1</font> implies <font color="Maroon" title="c10">pl1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">ql1</font> ) &amp; ( <font color="Maroon" title="c10">pl1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">ql1</font> implies <font color="Maroon" title="c8">p1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">q1</font> ) )
 <b>by </b><i><a class="txt" href="knaster.html#E1:31_1_2"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">p2</font> = <font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c13">q2</font> = <font color="Maroon" title="c9">q1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font> <b>by </b><i><a class="txt" href="knaster.html#E1:31_1_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="knaster.html#E2:31_1_2"><i><font color="Green" title="E23">A20</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">pl2</font>, <font color="Maroon" title="c15">ql2</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="E9:31_1_2_1_1"/><i><font color="Green" title="E30">A27</font></i>: 
( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pl2</font> &amp; <font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ql2</font> &amp; ( <font color="Maroon" title="c12">p2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">q2</font> implies <font color="Maroon" title="c14">pl2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">ql2</font> ) &amp; ( <font color="Maroon" title="c14">pl2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">ql2</font> implies <font color="Maroon" title="c12">p2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">q2</font> ) )
 <b>by </b><i><a class="txt" href="knaster.html#E2:31_1_2"><i><font color="Green" title="E23">A20</font></i></a></i>;<br/><b>thus </b><a NAME="E10:31_1_2_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c4">it2</font>
 <b>by </b><i><a class="txt" href="knaster.html#E6:31_1_2"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="knaster.html#E3:31_1_2_1_1"><i><font color="Green" title="E28">A25</font></i></a>, <a class="txt" href="knaster.html#E6:31_1_2_1_1"><i><font color="Green" title="E29">A26</font></i></a>, <a class="txt" href="knaster.html#E9:31_1_2_1_1"><i><font color="Green" title="E30">A27</font></i></a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:31_1_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c4">it2</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c7">q1</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font><b> such that </b><br/><a NAME="E4:31_1_2_1"/><i><font color="Green" title="E28">A28</font></i>: 
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">q1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c6">p1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c7">q1</font> )
 <b>by </b><i><a class="txt" href="knaster.html#E6:31_1_2"><i><font color="Green" title="E27">A24</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">p1</font> = <font color="Maroon" title="c6">p1</font>, <font color="Maroon" title="c9">q1</font> = <font color="Maroon" title="c7">q1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">it2</font> ;<br/><b>consider </b><font color="Maroon" title="c10">pl1</font>, <font color="Maroon" title="c11">ql1</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="E7:31_1_2_1"/><i><font color="Green" title="E29">A29</font></i>: 
( <font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">pl1</font> &amp; <font color="Maroon" title="c9">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">ql1</font> &amp; ( <font color="Maroon" title="c8">p1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">q1</font> implies <font color="Maroon" title="c10">pl1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">ql1</font> ) &amp; ( <font color="Maroon" title="c10">pl1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">ql1</font> implies <font color="Maroon" title="c8">p1</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c9">q1</font> ) )
 <b>by </b><i><a class="txt" href="knaster.html#E2:31_1_2"><i><font color="Green" title="E23">A20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">p2</font> = <font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c13">q2</font> = <font color="Maroon" title="c9">q1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">it1</font> <b>by </b><i><a class="txt" href="knaster.html#E1:31_1_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="knaster.html#E2:31_1_2"><i><font color="Green" title="E23">A20</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">pl2</font>, <font color="Maroon" title="c15">ql2</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="E10:31_1_2_1"/><i><font color="Green" title="E30">A30</font></i>: 
( <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">pl2</font> &amp; <font color="Maroon" title="c13">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">ql2</font> &amp; ( <font color="Maroon" title="c12">p2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">q2</font> implies <font color="Maroon" title="c14">pl2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">ql2</font> ) &amp; ( <font color="Maroon" title="c14">pl2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">ql2</font> implies <font color="Maroon" title="c12">p2</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">q2</font> ) )
 <b>by </b><i><a class="txt" href="knaster.html#E1:31_1_2"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/><b>thus </b><a NAME="E11:31_1_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c3">it1</font>
 <b>by </b><i><a class="txt" href="knaster.html#E5:31_1_2"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="knaster.html#E4:31_1_2_1"><i><font color="Green" title="E28">A28</font></i></a>, <a class="txt" href="knaster.html#E7:31_1_2_1"><i><font color="Green" title="E29">A29</font></i></a>, <a class="txt" href="knaster.html#E10:31_1_2_1"><i><font color="Green" title="E30">A30</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:31_1_2"/><b>then </b>
 <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c4">it2</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>
<b>hence </b><a NAME="E9:31_1_2"/>
<font color="Maroon" title="c3">it1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">it2</font>
 <b>by </b><i><a class="txt" href="knaster.html#E1:31_1_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="knaster.html#E2:31_1_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="txt" href="knaster.html#E3:31_1_2"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="knaster.html#E4:31_1_2"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="lattice3.html#T6" title="LATTICE3:th.6">LATTICE3:6</a></i>;<br/>


</div>
