
theorem Th19:
  for m be non zero Element of NAT, k be Element of NAT,
      X be non empty Subset of REAL m, f be PartFunc of REAL m,REAL,
      d be Real st X is open & f = X --> d
    holds
      f is_continuously_differentiable_up_to_order k,X
proof
  let m be non zero Element of NAT, k be Element of NAT,
      X be non empty Subset of REAL m, f be PartFunc of REAL m,REAL,
      d be Real;
  assume
A1: X is open & f = X --> d; then
  f is_partial_differentiable_up_to_order k,X by Th18;
  hence thesis by A1,Th18,FUNCT_2:def 1;
end;
