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 "\/" (("\/" F,(T |^ the carrier of S)) .: D),T = ("\/" F,(T |^ the carrier of S)) . (sup D)
let F be non empty Subset of (ContMaps S,T); :: thesis: for D being non empty directed Subset of S holds "\/" (("\/" F,(T |^ the carrier of S)) .: D),T = ("\/" F,(T |^ the carrier of S)) . (sup D)
let D be non empty directed Subset of S; :: thesis: "\/" (("\/" F,(T |^ the carrier of S)) .: D),T = ("\/" F,(T |^ the carrier of S)) . (sup D)
ContMaps S,T is full SubRelStr of T |^ the carrier of S by Def3;
then the carrier of (ContMaps S,T) c= the carrier of (T |^ the carrier of S) by YELLOW_0:def 13;
then reconsider F' = F as non empty Subset of (T |^ the carrier of S) by XBOOLE_1:1;
reconsider sF = sup F' as Function of S,T by Th19;
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;
deffunc H₁( 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 H₂( Element of (T |^ the carrier of S)) -> set = $1 . (sup D);
defpred S₁[ set ] means $1 in F';
A1: for g8 being Element of (T |^ the carrier of S) st S₁[g8] holds
H₁(g8) = H₂(g8)

proof

let g1 be Element of (T |^ the carrier of S); :: thesis: ( S₁[g1] implies H₁(g1) = H₂(g1) )
assume g1 in F' ; :: thesis: H₁(g1) = H₂(g1)
then reconsider g' = g1 as continuous Function of S,T by Th21;
A2: g' preserves_sup_of D by WAYBEL_0:def 37;
A3: ex_sup_of D,S by YELLOW_0:17;
the carrier of S c= dom g' by FUNCT_2:def 1;
then A4: the carrier of S c= dom g1 ;
deffunc H₃( Element of S) -> Element of S = $1;
defpred S₂[ set ] means $1 in D;
g1 .: { H₃(i2) where i2 is Element of S : S₂[i2] } = { (g1 . H₃(i)) where i is Element of S : S₂[i] } from WAYBEL24:sch 5(A4);
then "\/" { (g1 . i) where i is Element of S : i in D } ,T = sup (g' .: D) by Lm1
.= g1 . (sup D) by A2, A3, WAYBEL_0:def 31 ;
hence H₁(g1) = H₂(g1) ; :: thesis: verum

end;

{ H₁(g3) where g3 is Element of (T |^ the carrier of S) : S₁[g3] } = { H₂(g4) where g4 is Element of (T |^ the carrier of S) : S₁[g4] } from WAYBEL24:sch 4(A1);
then A5: "\/" { ("\/" { (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 = sF . (sup D) by Th25;
"\/" { ("\/" { (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 = sup (sF .: D) by Th27;
hence "\/" (("\/" F,(T |^ the carrier of S)) .: D),T = ("\/" F,(T |^ the carrier of S)) . (sup D) by A5, Th30; :: thesis: verum