
theorem Th14:
  for m be non zero Element of NAT, f be PartFunc of REAL m,REAL,
      X be non empty Subset of REAL m, d be Real
        st X is open & f = X --> d
    holds
      f is_differentiable_on X & dom (f`|X) = X
        & for x be Element of REAL m st x in X
          holds (f`|X)/.x = (REAL m --> 0)
proof
  let m be non zero Element of NAT, f be PartFunc of REAL m,REAL,
      X be non empty Subset of REAL m, d be Real;
  assume
A1: X is open & f = X --> d;
A2:X = dom f by A1,FUNCT_2:def 1;
  for x be Element of REAL m st x in X holds f is_differentiable_in x
                                                        by Th12,A1;
  hence f is_differentiable_on X by A2,A1,PDIFF_9:54;
  thus dom (f`|X) = X by PDIFF_9:def 4,A2;
  thus for x be Element of REAL m st x in X holds (f`|X)/.x = REAL m --> 0
  proof
    let x be Element of REAL m;
    assume
A3:   x in X;
    thus (f`|X)/.x = diff(f,x) by A2,PDIFF_9:def 4,A3
             .= REAL m --> 0 by A3,Th12,A1;
  end;
end;
