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
reconsider sF = "\/" F,(T |^ the carrier of S) as Function of S,T by Th19;
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 ;
defpred S1[ set ] means $1 in F;
deffunc H1( Element of S) -> Element of S = $1;
defpred S2[ set ] means $1 in D;
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);
A1: P = sup (sF .: D) by Th28;
{ ("\/" { (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 and
A3: g' in F ;
reconsider g1 = g' as continuous Function of S,T by A3, Th21;
g' <= "\/" F,(T |^ the carrier of S) by A3, YELLOW_2:24;
then A4: g1 <= sF by WAYBEL10:12;
A5: 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 and
A7: x in D and
A8: a = g1 . x by FUNCT_1:def 12;
reconsider x' = x as Element of S by A6;
x in the carrier of S by A6;
then x' in dom sF by FUNCT_2:def 1;
then sF . x' in sF .: D by A7, FUNCT_1:def 12;
then A9: sF . x' <= sup (sF .: D) by YELLOW_2:24;
g1 . x' <= sF . x' by A4, YELLOW_2:10;
hence a <= sup (sF .: D) by A8, A9, YELLOW_0:def 2; :: thesis: verum
end;
the carrier of S c= dom g1 by FUNCT_2:def 1;
then A10: the carrier of S c= dom g' ;
g' .: { H1(i2) where i2 is Element of S : S2[i2] } = { (g' . H1(i)) where i is Element of S : S2[i] } from WAYBEL24:sch 5(A10);
 then b = "\/" (g' .: D),T by A2, Lm1;
hence b <= P by A1, A5, YELLOW_0:32; :: thesis: verum
end;
then A11: L <= P by YELLOW_0:32;
A12: for g8 being Element of (T |^ the carrier of S) st S1[g8] holds
H2(g8) = H3(g8)
proof
let g1 be Element of (T |^ the carrier of S); :: thesis: ( S1[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;
A13: ( g' preserves_sup_of D & ex_sup_of D,S ) by WAYBEL_0:def 37, YELLOW_0:17;
the carrier of S c= dom g' by FUNCT_2:def 1;
then A14: the carrier of S c= dom g1 ;
g1 .: { H1(i2) where i2 is Element of S : S2[i2] } = { (g1 . H1(i)) where i is Element of S : S2[i] } from WAYBEL24:sch 5(A14);
 then "\/" { (g1 . i) where i is Element of S : i in D } ,T = sup (g' .: D) by Lm1
.= g1 . (sup D) by A13, WAYBEL_0:def 31 ;
hence H2(g1) = H3(g1) ; :: thesis: verum
end;
{ H2(g3) where g3 is Element of (T |^ the carrier of S) : S1[g3] } = { H3(g4) where g4 is Element of (T |^ the carrier of S) : S1[g4] } from WAYBEL24:sch 4(A12);
 then A15: 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
A16: b = "\/" { (g' . i') where g' is Element of (T |^ the carrier of S) : g' in F } ,T and
A17: i' in D ;
i' in the carrier of S ;
then A18: i' in dom sF by FUNCT_2:def 1;
b = sF . i' by A16, Th26;
then b in sF .: D by A17, A18, FUNCT_1:def 12;
then A19: b <= sup (sF .: D) by YELLOW_2:24;
sF is monotone by Th29;
then sup (sF .: D) <= L by A15, WAYBEL17:15;
hence b <= L by A19, 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 A11, YELLOW_0:def 3; :: thesis: verum