
theorem Th45:
  for R,S,T being complete LATTICE for f being
  directed-sups-preserving Function of R, UPS(S, T) holds uncurry f is
  directed-sups-preserving Function of [:R,S:], T
proof
  let R,S,T be complete LATTICE;
  let f be directed-sups-preserving Function of R, UPS(S, T);
A1: f in Funcs(the carrier of R, the carrier of UPS(S, T)) by FUNCT_2:8;
  Funcs(the carrier of R, the carrier of UPS(S, T)) c= Funcs(the carrier
  of R, Funcs(the carrier of S, the carrier of T)) by Th22,FUNCT_5:56;
  then uncurry f in Funcs([:the carrier of R, the carrier of S:], the carrier
  of T) by A1,FUNCT_6:11;
  then uncurry f in Funcs(the carrier of [:R, S:], the carrier of T) by
YELLOW_3:def 2;
  then reconsider g = uncurry f as Function of [:R, S:], T by FUNCT_2:66;
A2: the carrier of UPS(S, T) c= Funcs(the carrier of S, the carrier of T) by
Th22;
  now
    reconsider ST = UPS(S, T) as full sups-inheriting non empty SubRelStr of T
    |^the carrier of S by Def4,Th26;
    let a be Element of R, b be Element of S;
    reconsider f9 = f as directed-sups-preserving Function of R, ST;
    incl(ST, T|^the carrier of S) is directed-sups-preserving by WAYBEL21:10;
    then
A3: incl(ST, T|^the carrier of S)*f9 is directed-sups-preserving by WAYBEL20:28
;
    the carrier of ST c= the carrier of T|^the carrier of S by YELLOW_0:def 13;
    then incl(ST, T|^the carrier of S) = id the carrier of ST by YELLOW_9:def 1
;
    then f is directed-sups-preserving Function of R, T|^the carrier of S by A3
,FUNCT_2:17;
    then
A4: (commute f).b is directed-sups-preserving Function of R, T by Th38;
    rng f c= Funcs(the carrier of S, the carrier of T) by A2;
    then curry g = f by FUNCT_5:48;
    then Proj(g,a) = f.a by WAYBEL24:def 1;
    hence Proj (g, a) is directed-sups-preserving by Def4;
    Proj (g, b) = (curry' g).b by WAYBEL24:def 2;
    hence Proj (g, b) is directed-sups-preserving by A4,FUNCT_6:def 10;
  end;
  hence thesis by WAYBEL24:18;
end;
