
theorem Th40:
  for X being non empty set, S,T being non empty Poset for f being
  Function of X, the carrier of UPS(S, T) holds commute f is
  directed-sups-preserving Function of S, T|^X
proof
  let X be non empty set, S,T be non empty Poset;
  let f be Function of X, the carrier of UPS(S, T);
A1: the carrier of T|^X = Funcs(X, the carrier of T) by YELLOW_1:28;
A2: f in Funcs(X, the carrier of UPS(S, T)) by FUNCT_2:8;
A3: Funcs(X, the carrier of UPS(S, T)) c= Funcs(X, Funcs(the carrier of S,
  the carrier of T)) by Th22,FUNCT_5:56;
  then
  commute f in Funcs(the carrier of S, Funcs(X, the carrier of T)) by A2,
FUNCT_6:55;
  then reconsider g = commute f as Function of S, T|^X by A1,FUNCT_2:66;
A4: rng g c= Funcs(X, the carrier of T) by A1;
  g is directed-sups-preserving
  proof
    let A be Subset of S;
    assume A is non empty directed;
    then reconsider B = A as directed non empty Subset of S;
A5: T|^X = product (X --> T) by YELLOW_1:def 5;
    then
A6: dom (g.sup A) = X by WAYBEL_3:27;
    assume
A7: ex_sup_of A, S;
    now
      let x be Element of X;
      reconsider fx = f.x as directed-sups-preserving Function of S, T by Def4;
A8:   fx preserves_sup_of B by WAYBEL_0:def 37;
      commute g = f by A3,A2,FUNCT_6:57;
      then
A9:   fx.:A = pi(g.:A, x) by A4,Th8;
      thus ex_sup_of pi(g.:A, x), (X --> T).x by A9,A8,A7;
    end;
    hence
A10: ex_sup_of g.:A, T|^X by A5,YELLOW16:31;
A11: now
      let x be object;
      assume x in X;
      then reconsider a = x as Element of X;
      reconsider fx = f.a as directed-sups-preserving Function of S, T by Def4;
A12:  (X --> T).a = T;
      commute g = f by A3,A2,FUNCT_6:57;
      then
A13:  fx.:A = pi(g.:A, a) by A4,Th8;
      fx preserves_sup_of B by WAYBEL_0:def 37;
      then sup pi(g.:B, a) = fx.sup A by A13,A7;
      then fx.sup A = (sup (g.:B)).a by A5,A10,A12,YELLOW16:33;
      hence (sup (g.:A)).x = (g.sup A).x by A3,A2,FUNCT_6:56;
    end;
    dom sup (g.:A) = X by A5,WAYBEL_3:27;
    hence thesis by A11,A6;
  end;
  hence thesis;
end;
