
theorem Th16:
  for S, T being non empty reflexive antisymmetric RelStr,
  D being directed non empty Subset of S, f being Function of S, T st
  ex_sup_of D,S & ex_sup_of f.:D,T or
  S is up-complete & T is up-complete holds
  f is monotone implies sup (f.:D) <= f.(sup D)
proof
  let S, T be non empty reflexive antisymmetric RelStr;
  let D be directed non empty Subset of S;
  let f be Function of S, T;
  assume that
A1: ex_sup_of D,S & ex_sup_of f.:D,T or S is up-complete & T is up-complete;
  assume
A2: f is monotone;
  then reconsider fD = f.:D as directed non empty Subset of T by YELLOW_2:15;
A3: ex_sup_of fD, T by A1,WAYBEL_0:75;
  ex_sup_of D, S by A1,WAYBEL_0:75;
  then D is_<=_than sup D by YELLOW_0:30;
  then f.:D is_<=_than f.(sup D) by A2,YELLOW_2:14;
  hence thesis by A3,YELLOW_0:30;
end;
