let S, T be complete Scott TopLattice; :: thesis: for F being non empty Subset of (ContMaps S,T)
for D being non empty directed Subset of S holds "\/" { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ,T = "\/" { ("\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T) where i' is Element of S : i' in D } ,T
let F be non empty Subset of (ContMaps S,T); :: thesis: for D being non empty directed Subset of S holds "\/" { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ,T = "\/" { ("\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T) where i' is Element of S : i' in D } ,T
let D be non empty directed Subset of S; :: thesis: "\/" { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ,T = "\/" { ("\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T) where i' is Element of S : i' in D } ,T
set F' = F;
set L = "\/" { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ,T;
set P = "\/" { ("\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T) where i' is Element of S : i' in D } ,T;
set L1 = { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ;
set P1 = { ("\/" { (g2 . i3) where g2 is Element of (T |^ the carrier of S) : g2 in F } ,T) where i3 is Element of S : i3 in D } ;
reconsider L = "\/" { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ,T, P = "\/" { ("\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T) where i' is Element of S : i' in D } ,T as Element of T ;
reconsider sF = "\/" F,(T |^ the carrier of S) as Function of S,T by Th19;
A1: P = sup (sF .: D) by Th28;
deffunc H₁( Element of S) -> Element of S = $1;
defpred S₁[ set ] means $1 in D;
{ ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } is_<=_than P

proof

let b be Element of T; :: according to LATTICE3:def 9 :: thesis: ( not b in { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } or b <= P )
assume b in { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ; :: thesis: b <= P
then consider g' being Element of (T |^ the carrier of S) such that
A2: ( b = "\/" { (g' . i) where i is Element of S : i in D } ,T & g' in F ) ;
reconsider g1 = g' as continuous Function of S,T by A2, Th21;
the carrier of S c= dom g1 by FUNCT_2:def 1;
then A3: the carrier of S c= dom g' ;
g' .: { H₁(i2) where i2 is Element of S : S₁[i2] } = { (g' . H₁(i)) where i is Element of S : S₁[i] } from WAYBEL24:sch 5(A3);
then A4: b = "\/" (g' .: D),T by A2, Lm1;
g' <= "\/" F,(T |^ the carrier of S) by A2, YELLOW_2:24;
then A5: g1 <= sF by WAYBEL10:12;
g1 .: D is_<=_than sup (sF .: D)

proof

let a be Element of T; :: according to LATTICE3:def 9 :: thesis: ( not a in g1 .: D or a <= sup (sF .: D) )
assume a in g1 .: D ; :: thesis: a <= sup (sF .: D)
then consider x being set such that
A6: ( x in dom g1 & x in D & a = g1 . x ) by FUNCT_1:def 12;
A7: x in the carrier of S by A6;
reconsider x' = x as Element of S by A6;
A8: g1 . x' <= sF . x' by A5, YELLOW_2:10;
x' in dom sF by A7, FUNCT_2:def 1;
then sF . x' in sF .: D by A6, FUNCT_1:def 12;
then sF . x' <= sup (sF .: D) by YELLOW_2:24;
hence a <= sup (sF .: D) by A6, A8, YELLOW_0:def 2; :: thesis: verum

end;

hence b <= P by A1, A4, YELLOW_0:32; :: thesis: verum

end;

then A9: L <= P by YELLOW_0:32;
deffunc H₂( Element of (T |^ the carrier of S)) -> Element of the carrier of T = "\/" { ($1 . i4) where i4 is Element of S : i4 in D } ,T;
deffunc H₃( Element of (T |^ the carrier of S)) -> set = $1 . (sup D);
defpred S₂[ set ] means $1 in F;
A10: for g8 being Element of (T |^ the carrier of S) st S₂[g8] holds
H₂(g8) = H₃(g8)

proof

let g1 be Element of (T |^ the carrier of S); :: thesis: ( S₂[g1] implies H₂(g1) = H₃(g1) )
assume g1 in F ; :: thesis: H₂(g1) = H₃(g1)
then reconsider g' = g1 as continuous Function of S,T by Th21;
A11: g' preserves_sup_of D by WAYBEL_0:def 37;
A12: ex_sup_of D,S by YELLOW_0:17;
the carrier of S c= dom g' by FUNCT_2:def 1;
then A13: the carrier of S c= dom g1 ;
g1 .: { H₁(i2) where i2 is Element of S : S₁[i2] } = { (g1 . H₁(i)) where i is Element of S : S₁[i] } from WAYBEL24:sch 5(A13);
then "\/" { (g1 . i) where i is Element of S : i in D } ,T = sup (g' .: D) by Lm1
.= g1 . (sup D) by A11, A12, WAYBEL_0:def 31 ;
hence H₂(g1) = H₃(g1) ; :: thesis: verum

end;

{ H₂(g3) where g3 is Element of (T |^ the carrier of S) : S₂[g3] } = { H₃(g4) where g4 is Element of (T |^ the carrier of S) : S₂[g4] } from WAYBEL24:sch 4(A10);
then A14: L = sF . (sup D) by Th26;
{ ("\/" { (g2 . i3) where g2 is Element of (T |^ the carrier of S) : g2 in F } ,T) where i3 is Element of S : i3 in D } is_<=_than L

proof

let b be Element of T; :: according to LATTICE3:def 9 :: thesis: ( not b in { ("\/" { (g2 . i3) where g2 is Element of (T |^ the carrier of S) : g2 in F } ,T) where i3 is Element of S : i3 in D } or b <= L )
assume b in { ("\/" { (g2 . i3) where g2 is Element of (T |^ the carrier of S) : g2 in F } ,T) where i3 is Element of S : i3 in D } ; :: thesis: b <= L
then consider i' being Element of S such that
A15: ( b = "\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T & i' in D ) ;
A16: b = sF . i' by A15, Th26;
sF is monotone by Th29;
then A17: sup (sF .: D) <= L by A14, WAYBEL17:15;
i' in the carrier of S ;
then i' in dom sF by FUNCT_2:def 1;
then b in sF .: D by A15, A16, FUNCT_1:def 12;
then b <= sup (sF .: D) by YELLOW_2:24;
hence b <= L by A17, YELLOW_0:def 2; :: thesis: verum

end;

then P <= L by YELLOW_0:32;
hence "\/" { ("\/" { (g . i) where i is Element of S : i in D } ,T) where g is Element of (T |^ the carrier of S) : g in F } ,T = "\/" { ("\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T) where i' is Element of S : i' in D } ,T by A9, YELLOW_0:def 3; :: thesis: verum