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

<b>set </b><font color="Maroon" title="c3">LL</font> =  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c4">cL</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>;<br/>
<b>set </b><font color="Maroon" title="c5">LR</font> = <span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">P</font>;<br/>
<a NAME="E1:31_1_1"/>
<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">LR</font> = <span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="wellord1.html#K2" title="WELLORD1:func.2">|_2</a> <font color="Maroon" title="c2">P</font> as   <a href="relset_1.html#NM3" title="RELSET_1:NM.3">Relation</a> of <font color="Maroon" title="c2">P</font> <b>by </b><i><a class="ref" href="relset_1.html#D1" title="RELSET_1:def.1">RELSET_1:def 1</a></i>;<br/>
<a NAME="E3:31_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T33" title="FILTER_1:th.33">FILTER_1:33</a></i>;<br/>
<a NAME="E4:31_1_1"/><b>then </b><i><font color="Green" title="E22">A1</font></i>: 
(  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> &amp;  <a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c1">L</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E5:31_1_1"/><i><font color="Green" title="E23">A2</font></i>: 
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">LR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_1_1_1"/>
 <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">LR</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="ref" href="wellord1.html#T20" title="WELLORD1:th.20">WELLORD1:20</a></i>; <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><br/>

<b>let </b><font color="Maroon" title="c7">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><br/>

<b>assume </b><a NAME="E2:31_1_1_1"/><i><font color="Green" title="E24">A3</font></i>: 
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 ;<br/>

<a NAME="E3:31_1_1_1"/><b>then </b><i><font color="Green" title="E25">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E4:31_1_1_1"/><i><font color="Green" title="E26">A5</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="txt" href="knaster.html#E2:31_1_1_1"><i><font color="Green" title="E24">A3</font></i></a>, <a class="ref" href="relat_2.html#D1" title="RELAT_2:def.1">RELAT_2:def 1</a></i>;<br/>
<a NAME="E5:31_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c7">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">LR</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E3:31_1_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="knaster.html#E4:31_1_1_1"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E6:31_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K3" title="RELAT_1:func.3">field</a> <font color="Maroon" title="c6">LR</font>
 <b>by </b><i><a class="ref" href="relat_1.html#T30" title="RELAT_1:th.30">RELAT_1:30</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:31_1_1"/>
<font color="Maroon" title="c6">LR</font> is <a href="relat_2.html#V1" title="RELAT_2:attr.1">reflexive</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T22" title="WELLORD1:th.22">WELLORD1:22</a></i>;<br/>
<a NAME="E7:31_1_1"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c6">LR</font> <a href="relat_2.html#R1" title="RELAT_2:pred.1">is_reflexive_in</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E5:31_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="relat_2.html#D9" title="RELAT_2:def.9">RELAT_2:def 9</a></i>;<br/>
<a NAME="E8:31_1_1"/>
<font color="Maroon" title="c6">LR</font> is <a href="relat_2.html#V4" title="RELAT_2:attr.4">antisymmetric</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T25" title="WELLORD1:th.25">WELLORD1:25</a></i>;<br/>
<a NAME="E9:31_1_1"/><b>then </b><i><font color="Green" title="E25">A7</font></i>: 
<font color="Maroon" title="c6">LR</font> <a href="relat_2.html#R4" title="RELAT_2:pred.4">is_antisymmetric_in</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E5:31_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="relat_2.html#D12" title="RELAT_2:def.12">RELAT_2:def 12</a></i>;<br/>
<a NAME="E10:31_1_1"/>
<font color="Maroon" title="c6">LR</font> is <a href="relat_2.html#V8" title="RELAT_2:attr.8">transitive</a>
 
<b>by </b><i><a class="ref" href="wellord1.html#T24" title="WELLORD1:th.24">WELLORD1:24</a></i>;<br/>
<a NAME="E11:31_1_1"/><b>then </b><i><font color="Green" title="E26">A8</font></i>: 
<font color="Maroon" title="c6">LR</font> <a href="relat_2.html#R8" title="RELAT_2:pred.8">is_transitive_in</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E5:31_1_1"><i><font color="Green" title="E23">A2</font></i></a>, <a class="ref" href="relat_2.html#D16" title="RELAT_2:def.16">RELAT_2:def 16</a></i>;<br/>
<a NAME="E12:31_1_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">LR</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E7:31_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="orders_1.html#T98" title="ORDERS_1:th.98">ORDERS_1:98</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">LR</font> = <font color="Maroon" title="c6">LR</font> as   <a href="orders_1.html#NM2" title="ORDERS_1:NM.2">Order</a> of <font color="Maroon" title="c2">P</font> <b>by </b><i><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="wellord1.html#T22" title="WELLORD1:th.22">WELLORD1:22</a>, <a class="ref" href="wellord1.html#T24" title="WELLORD1:th.24">WELLORD1:24</a>, <a class="ref" href="wellord1.html#T25" title="WELLORD1:th.25">WELLORD1:25</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">RS</font> =  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #);<br/>
<a NAME="E14:31_1_1"/><i><font color="Green" title="E27">A9</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #) is <a href="lattice3.html#V1" title="LATTICE3:attr.1">with_suprema</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D10" title="LATTICE3:def.10">LATTICE3:def 10</a><br/>

<a NAME="E1:31_1_1_2"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">xL</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">yL</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>consider </b><font color="Maroon" title="c13">zL</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="E4:31_1_1_2"/><i><font color="Green" title="E28">A10</font></i>: 
( <font color="Maroon" title="c13">zL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c11">xL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">zL</font> &amp; <font color="Maroon" title="c12">yL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">zL</font> &amp; ( for <font color="Olive" title="b1">z'</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="Olive" title="b1">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c11">xL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> &amp; <font color="Maroon" title="c12">yL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> holds <br/><font color="Maroon" title="c13">zL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> ) )
 <b>by </b><i><a class="ref" href="knaster.html#D8" target="_self" title="KNASTER:def.8">Def8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c13">zL</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #) <b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">z</font>
;<br/>

<a NAME="E6:31_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">xL</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">yL</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">xL</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">yL</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:31_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">xL</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">yL</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_1_2"/>
( <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z</font> &amp; <font color="Maroon" title="c10">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #);<br/>

<b>assume </b><a NAME="E9:31_1_1_2"/><i><font color="Green" title="E29">A11</font></i>: 
( <font color="Maroon" title="c9">x</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">z'</font> &amp; <font color="Maroon" title="c10">y</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">z'</font> )
 ;<br/>

<a NAME="E10:31_1_1_2"/>
<font color="Maroon" title="c15">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">z'L</font> = <font color="Maroon" title="c15">z'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E12:31_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c15">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c15">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E9:31_1_1_2"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E13:31_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c15">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c15">z'</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:31_1_1_2"/><b>then </b>
( <font color="Maroon" title="c11">xL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c16">z'L</font> &amp; <font color="Maroon" title="c12">yL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c16">z'L</font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/>
<a NAME="E15:31_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">zL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c16">z'L</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/>
<a NAME="E16:31_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c16">z'L</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c16">z'L</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_2"><i><font color="Green" title="E28">A10</font></i></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:31_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c16">z'L</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E18:31_1_1_2"/>
<font color="Maroon" title="c14">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c15">z'</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:31_1_1"/><i><font color="Green" title="E28">A12</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #) is <a href="lattice3.html#V2" title="LATTICE3:attr.2">with_infima</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c10">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #); <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D11" title="LATTICE3:def.11">LATTICE3:def 11</a><br/>

<a NAME="E1:31_1_1_3"/>
( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">xL</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c12">yL</font> = <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>consider </b><font color="Maroon" title="c13">zL</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="E4:31_1_1_3"/><i><font color="Green" title="E29">A13</font></i>: 
( <font color="Maroon" title="c13">zL</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c13">zL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">xL</font> &amp; <font color="Maroon" title="c13">zL</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">yL</font> &amp; ( for <font color="Olive" title="b1">z'</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="Olive" title="b1">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Olive" title="b1">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">xL</font> &amp; <font color="Olive" title="b1">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">yL</font> holds <br/><font color="Olive" title="b1">z'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">zL</font> ) )
 <b>by </b><i><a class="ref" href="knaster.html#D9" target="_self" title="KNASTER:def.9">Def9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">z</font> = <font color="Maroon" title="c13">zL</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #) <b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green" title="E29">A13</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c14">z</font>
;<br/>

<a NAME="E6:31_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c11">xL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c12">yL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c11">xL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c12">yL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:31_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c11">xL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">zL</font>,<font color="Maroon" title="c12">yL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:31_1_1_3"/>
( <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c14">z</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">y</font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>

<b>let </b><font color="Maroon" title="c15">z'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #);<br/>

<b>assume </b><a NAME="E9:31_1_1_3"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c15">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon" title="c15">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">y</font> )
 ;<br/>

<a NAME="E10:31_1_1_3"/>
<font color="Maroon" title="c15">z'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">z'L</font> = <font color="Maroon" title="c15">z'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<a NAME="E12:31_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">z'</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">z'</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E9:31_1_1_3"><i><font color="Green" title="E30">A14</font></i></a>, <a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E13:31_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">z'</font>,<font color="Maroon" title="c9">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">z'</font>,<font color="Maroon" title="c10">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E14:31_1_1_3"/><b>then </b>
( <font color="Maroon" title="c16">z'L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">xL</font> &amp; <font color="Maroon" title="c16">z'L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c12">yL</font> )
 
<b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/>
<a NAME="E15:31_1_1_3"/><b>then </b>
<font color="Maroon" title="c16">z'L</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">zL</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green" title="E29">A13</font></i></a></i>;<br/>
<a NAME="E16:31_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z'L</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z'L</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E4:31_1_1_3"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E17:31_1_1_3"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z'L</font>,<font color="Maroon" title="c13">zL</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E18:31_1_1_3"/>
<font color="Maroon" title="c15">z'</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c14">z</font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9" title="ORDERS_2:def.9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E16:31_1_1"/><i><font color="Green" title="E29">A15</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #) is   <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 
<b>by </b><i><a class="txt" href="knaster.html#E7:31_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="txt" href="knaster.html#E9:31_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="txt" href="knaster.html#E11:31_1_1"><i><font color="Green" title="E26">A8</font></i></a>, <a class="ref" href="orders_2.html#D4" title="ORDERS_2:def.4">ORDERS_2:def 4</a>, <a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a>, <a class="ref" href="orders_2.html#D6" title="ORDERS_2:def.6">ORDERS_2:def 6</a></i>;<br/>
<b>take </b><font color="Maroon" title="c9">IT</font> =  <a href="lattice3.html#K14" title="LATTICE3:func.14">latt</a> <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #);<br/>

<a NAME="E17:31_1_1"/><i><font color="Green" title="E30">A16</font></i>: 
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c9">IT</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="c9">IT</font>,<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <font color="Maroon" title="c9">IT</font></span>)</span> #)
 
;<br/>
<a NAME="E18:31_1_1"/><i><font color="Green" title="E31">A17</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c7">LR</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <font color="Maroon" title="c9">IT</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E14:31_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="knaster.html#E15:31_1_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="knaster.html#E16:31_1_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="lattice3.html#D15" title="LATTICE3:def.15">LATTICE3:def 15</a></i>;<br/>
<b>thus </b><a NAME="E19:31_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c9">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font>
 <b>by </b><i><a class="txt" href="knaster.html#E14:31_1_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="knaster.html#E15:31_1_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="knaster.html#E16:31_1_1"><i><font color="Green" title="E29">A15</font></i></a>, <a class="txt" href="knaster.html#E17:31_1_1"><i><font color="Green" title="E30">A16</font></i></a>, <a class="ref" href="lattice3.html#D15" title="LATTICE3:def.15">LATTICE3:def 15</a></i>;<br/>

<b>let </b><font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c11">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c9">IT</font>;<br/>

<a NAME="E20:31_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> )
 
<b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green" title="E31">A17</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">x'</font> = <font color="Maroon" title="c10">x</font>, <font color="Maroon" title="c13">y'</font> = <font color="Maroon" title="c11">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c12">x'</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c13">y'</font>
;<br/>

<b>thus </b><a NAME="E22:31_1_1"/>
( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">x'</font> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y'</font> )
 ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:31_1_1_4"/>
<font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font>
 ;<br/><a NAME="E2:31_1_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">LR</font>
 <b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/><a NAME="E3:31_1_1_4"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>hence </b><a NAME="E4:31_1_1_4"/>
<font color="Maroon" title="c12">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">y'</font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E24:31_1_1"/><i><font color="Green" title="E32">A18</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c2">P</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>assume </b><a NAME="E25:31_1_1"/>
<font color="Maroon" title="c12">x'</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">y'</font>
 ;<br/>

<a NAME="E26:31_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c1">L</font>
 
<b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/>
<a NAME="E27:31_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c11">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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="c9">IT</font>
 
<b>by </b><i><a class="txt" href="knaster.html#E18:31_1_1"><i><font color="Green" title="E31">A17</font></i></a>, <a class="txt" href="knaster.html#E24:31_1_1"><i><font color="Green" title="E32">A18</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E28:31_1_1"/>
<font color="Maroon" title="c10">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c11">y</font>
 <b>by </b><i><a class="ref" href="filter_1.html#T32" title="FILTER_1:th.32">FILTER_1:32</a></i>;<br/>


</div>
