
theorem Th39:
  for X being non empty set, S,T being non empty Poset for f being
directed-sups-preserving Function of S, T|^X holds commute f is Function of X,
  the carrier of UPS(S, T)
proof
  let M be non empty set, X,Y be non empty Poset;
  let f be directed-sups-preserving Function of X, Y|^M;
A1: rng commute f c= the carrier of UPS(X, Y)
  proof
    let g be object;
    assume g in rng commute f;
    then consider i being object such that
A2: i in dom commute f and
A3: g = (commute f).i by FUNCT_1:def 3;
    reconsider i as Element of M by A2,Lm1;
    (commute f).i is directed-sups-preserving Function of X, Y by Th38;
    hence thesis by A3,Def4;
  end;
  dom commute f = M by Lm1;
  hence thesis by A1,FUNCT_2:2;
end;
