reserve m, n for non zero Element of NAT;
reserve i, j, k for Element of NAT;
reserve Z for Subset of REAL 2;
reserve c for Real;
reserve I for non empty FinSequence of NAT;
reserve d1, d2 for Element of REAL;

theorem LM003:
  for f be PartFunc of REAL, REAL, X be Subset of REAL
  st X is open & X c= dom f
  holds f is_differentiable_on X iff (f|X) is_differentiable_on X
  proof
    let f be PartFunc of REAL, REAL, X be Subset of REAL;
    assume that
    AS1: X is open and
    AS2: X c= dom f;
    hereby
      assume
      A1: f is_differentiable_on X;
      A3: dom(f|X) = X by RELAT_1:62, AS2;
      now
        let x be Real;
        assume
        A4: x in X;
        then f is_differentiable_in x by A1, AS1, FDIFF_1:9;
        hence (f|X) is_differentiable_in x by LM002, AS1, A4;
      end;
      hence (f|X) is_differentiable_on X by AS1, A3, FDIFF_1:9;
    end;
    assume
    A1: (f|X) is_differentiable_on X;
    now
      let x be Real;
      assume
      A4: x in X;
      then (f|X) is_differentiable_in x by A1, AS1, FDIFF_1:9;
      hence f is_differentiable_in x by LM002, AS1, A4;
    end;
    hence f is_differentiable_on X by AS1, AS2, FDIFF_1:9;
  end;
