let D1, D2 be non empty set ; :: thesis: for f1, g1 being BinOp of D1
for f2, g2 being BinOp of D2 holds
( ( f1 absorbs g1 & f2 absorbs g2 ) iff |:f1,f2:| absorbs |:g1,g2:| )
let f1, g1 be BinOp of D1; :: thesis: for f2, g2 being BinOp of D2 holds
( ( f1 absorbs g1 & f2 absorbs g2 ) iff |:f1,f2:| absorbs |:g1,g2:| )
let f2, g2 be BinOp of D2; :: thesis: ( ( f1 absorbs g1 & f2 absorbs g2 ) iff |:f1,f2:| absorbs |:g1,g2:| )
defpred S1[ set , set ] means |:f1,f2:| . ($1,(|:g1,g2:| . ($1,$2))) = $1;
thus ( f1 absorbs g1 & f2 absorbs g2 implies |:f1,f2:| absorbs |:g1,g2:| ) :: thesis: ( |:f1,f2:| absorbs |:g1,g2:| implies ( f1 absorbs g1 & f2 absorbs g2 ) )
proof
assume A1: for a1, b1 being Element of D1 holds f1 . (a1,(g1 . (a1,b1))) = a1 ; :: according to LATTICE2:def 1 :: thesis: ( not f2 absorbs g2 or |:f1,f2:| absorbs |:g1,g2:| )
assume A2: for a2, b2 being Element of D2 holds f2 . (a2,(g2 . (a2,b2))) = a2 ; :: according to LATTICE2:def 1 :: thesis: |:f1,f2:| absorbs |:g1,g2:|
A3: for d1, d19 being Element of D1
for d2, d29 being Element of D2 holds S1[[d1,d2],[d19,d29]]
proof
let a1, b1 be Element of D1; :: thesis: for d2, d29 being Element of D2 holds S1[[a1,d2],[b1,d29]]
let a2, b2 be Element of D2; :: thesis: S1[[a1,a2],[b1,b2]]
thus |:f1,f2:| . ([a1,a2],(|:g1,g2:| . ([a1,a2],[b1,b2]))) = |:f1,f2:| . ([a1,a2],[(g1 . (a1,b1)),(g2 . (a2,b2))]) by Th21
.= [(f1 . (a1,(g1 . (a1,b1)))),(f2 . (a2,(g2 . (a2,b2))))] by Th21
.= [a1,(f2 . (a2,(g2 . (a2,b2))))] by A1
.= [a1,a2] by A2 ; :: thesis: verum
end;
thus for a, b being Element of [:D1,D2:] holds S1[a,b] from FILTER_1:sch 5(A3); :: according to LATTICE2:def 1 :: thesis: verum
end;
assume A4: for a, b being Element of [:D1,D2:] holds |:f1,f2:| . (a,(|:g1,g2:| . (a,b))) = a ; :: according to LATTICE2:def 1 :: thesis: ( f1 absorbs g1 & f2 absorbs g2 )
thus for a1, b1 being Element of D1 holds f1 . (a1,(g1 . (a1,b1))) = a1 :: according to LATTICE2:def 1 :: thesis: f2 absorbs g2
proof
set a2 = the Element of D2;
let a1, b1 be Element of D1; :: thesis: f1 . (a1,(g1 . (a1,b1))) = a1
[a1, the Element of D2] = |:f1,f2:| . ([a1, the Element of D2],(|:g1,g2:| . ([a1, the Element of D2],[b1, the Element of D2]))) by A4
.= |:f1,f2:| . ([a1, the Element of D2],[(g1 . (a1,b1)),(g2 . ( the Element of D2, the Element of D2))]) by Th21
.= [(f1 . (a1,(g1 . (a1,b1)))),(f2 . ( the Element of D2,(g2 . ( the Element of D2, the Element of D2))))] by Th21 ;
hence f1 . (a1,(g1 . (a1,b1))) = a1 by XTUPLE_0:1; :: thesis: verum
end;
set a1 = the Element of D1;
let a2 be Element of D2; :: according to LATTICE2:def 1 :: thesis: for b1 being Element of D2 holds f2 . (a2,(g2 . (a2,b1))) = a2
let b2 be Element of D2; :: thesis: f2 . (a2,(g2 . (a2,b2))) = a2
[ the Element of D1,a2] = |:f1,f2:| . ([ the Element of D1,a2],(|:g1,g2:| . ([ the Element of D1,a2],[ the Element of D1,b2]))) by A4
.= |:f1,f2:| . ([ the Element of D1,a2],[(g1 . ( the Element of D1, the Element of D1)),(g2 . (a2,b2))]) by Th21
.= [(f1 . ( the Element of D1,(g1 . ( the Element of D1, the Element of D1)))),(f2 . (a2,(g2 . (a2,b2))))] by Th21 ;
hence f2 . (a2,(g2 . (a2,b2))) = a2 by XTUPLE_0:1; :: thesis: verum