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

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="conlat_1.html#NM1" title="CONLAT_1:NM.1">FormalContext</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="conlat_1.html#V3" title="CONLAT_1:attr.3">co-Galois</a> </span><br/>

<a NAME="E1:30"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D2" title="LATTICE3:def.2">LATTICE3:def 2</a></span>;<br/>
<a NAME="E2:30"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
 <a href="lattice3.html#K3" title="LATTICE3:func.3">LattPOSet</a> <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="lattice3.html#K2" title="LATTICE3:func.2">LattRel</a> <span class="p2">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D2" title="LATTICE3:def.2">LATTICE3:def 2</a></span>;<br/>
<a NAME="E3:30"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span><br/>  for <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b1">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> holds <br/><font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> implies <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:30_1"/>
<font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font></span><br/>

<a NAME="E2:30_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E3:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span></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> <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">x9</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">x</font> = <font color="Maroon" title="c5">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<a NAME="E8:30_1"/>
<font color="Maroon" title="c7">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E9:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">x9</font> = <font color="Maroon" title="c4">x9</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">y</font> = <font color="Maroon" title="c7">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<a NAME="E12:30_1"/>
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">y</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_1"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E13:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E14:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#T4" target="_self" title="CONLAT_1:th.4">Th4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">y</font> = <font color="Maroon" title="c9">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">y9</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">y9</font> = <font color="Maroon" title="c11">y9</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">y9</font> = <font color="Maroon" title="c12">y9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<a NAME="E19:30_1"/><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#T5" target="_self" title="CONLAT_1:th.5">Th5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">x</font> = <font color="Maroon" title="c6">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<a NAME="E22:30_1"/>
<font color="Maroon" title="c15">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c13">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:30_1"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E19:30_1"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E23:30_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c13">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>;<br/>
<a NAME="E24:30_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></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> <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E25:30_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:30_1"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:30"/>
 for <font color="Olive" title="b1">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span><br/>  for <font color="Olive" title="b2">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Olive" title="b1">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b2">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>  st <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Olive" title="b1">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> implies <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:30_2"/>
<font color="Maroon" title="c2">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span><br/>

<a NAME="E2:30_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>;<br/>
<a NAME="E3:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font></span>)</span></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> <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">y9</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c3">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c5">y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">x</font> = <font color="Maroon" title="c2">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<a NAME="E8:30_2"/>
<font color="Maroon" title="c7">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_2"><span class="lab"><font color="Green" title="E18">A8</font></span></a>, <a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E9:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A9</font></span>: 
<font color="Maroon" title="c7">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">y9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">y9</font> = <font color="Maroon" title="c4">y9</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c7">x</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<a NAME="E12:30_2"/>
 for <font color="Olive" title="b1">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">x</font> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_2"><span class="lab"><font color="Green" title="E19">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E13:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A10</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">y9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font>, <font color="Maroon" title="c11">y9</font> = <font color="Maroon" title="c8">y9</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> ;<br/>
<a NAME="E15:30_2"/><span class="lab"><font color="Green" title="E21">A11</font></span>: 
<span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">y9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:30_2"><span class="lab"><font color="Green" title="E20">A10</font></span></a>, <a class="ref" href="conlat_1.html#T3" target="_self" title="CONLAT_1:th.3">Th3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">x</font> = <font color="Maroon" title="c10">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c13">x9</font> = <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">x9</font> = <font color="Maroon" title="c13">x9</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="lattice3.html#D1" title="LATTICE3:def.1">LATTICE3:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c15">x9</font> = <font color="Maroon" title="c14">x9</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<a NAME="E20:30_2"/><span class="lab"><font color="Green" title="E22">A12</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K1" title="CONLAT_1:func.1">ObjectDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K2" title="CONLAT_1:func.2">AttributeDerivation</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="conlat_1.html#T6" target="_self" title="CONLAT_1:th.6">Th6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">y</font> = <font color="Maroon" title="c6">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span> ;<br/>
<a NAME="E22:30_2"/>
<font color="Maroon" title="c16">y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c15">x9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:30_2"><span class="lab"><font color="Green" title="E21">A11</font></span></a>, <a class="txt" href="#E20:30_2"><span class="lab"><font color="Green" title="E22">A12</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a>, <a class="ref" href="xboole_1.html#T12" title="XBOOLE_1:th.12">XBOOLE_1:12</a></span>;<br/>
<a NAME="E23:30_2"/><span class="kw">then </span>
<font color="Maroon" title="c16">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c15">x9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D3" title="LATTICES:def.3">LATTICES:def 3</a></span>;<br/>
<a NAME="E24:30_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></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> <span class="p1">(<span class="default"><a href="lattice3.html#K1" title="LATTICE3:func.1">BooleLatt</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_1.html#T31" title="FILTER_1:th.31">FILTER_1:31</a></span>;<br/>
<a NAME="E25:30_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">y</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">x</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_1.html#K3" title="YELLOW_1:func.3">BoolePoset</a>  the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">C</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="ref" href="yellow_1.html#D2" title="YELLOW_1:def.2">YELLOW_1:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:30_2"/>
<font color="Maroon" title="c3">y</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="orders_2.html#D5" title="ORDERS_2:def.5">ORDERS_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:30"/>
<span class="p1"><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><a href="conlat_1.html#K3" title="CONLAT_1:func.3">phi</a> <font color="Maroon" title="c1">C</font></span>)</span>,<span class="p2">(<span class="default"><a href="conlat_1.html#K4" title="CONLAT_1:func.4">psi</a> <font color="Maroon" title="c1">C</font></span>)</span></span><a href="waybel_1.html#K1" title="WAYBEL_1:func.1">]</a></span> is  <a href="conlat_1.html#V3" title="CONLAT_1:attr.3">co-Galois</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="conlat_1.html#T12" target="_self" title="CONLAT_1:th.12">Th13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
