
theorem Th15:
  for S, T being non empty RelStr, D being Subset of S,
  f being Function of S, T st ex_sup_of D,S & ex_sup_of f.:D,T or
  S is complete antisymmetric & T is complete antisymmetric holds
  f is monotone implies sup (f.:D) <= f.(sup D)
proof
  let S, T be non empty RelStr;
  let D be 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 complete antisymmetric & T is complete antisymmetric;
A2: ex_sup_of D,S by A1,YELLOW_0:17;
A3: ex_sup_of f.:D,T by A1,YELLOW_0:17;
  assume
A4: f is monotone;
  D is_<=_than sup D by A2,YELLOW_0:def 9;
  then f.:D is_<=_than f.(sup D) by A4,YELLOW_2:14;
  hence thesis by A3,YELLOW_0:def 9;
end;
