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 DIFF2:
  for f be PartFunc of REAL, REAL, Z be Subset of REAL
  st Z c= dom f & Z is open & f is_differentiable_on 2,Z holds
  f is_differentiable_on Z & ((diff(f,Z)).1) = f`| Z
  & f`| Z is_differentiable_on Z & ((diff(f,Z)).2) = (f`| Z)`| Z
  proof
    let f be PartFunc of REAL, REAL, Z be Subset of REAL;
    assume
    AS: Z c= dom f & Z is open & f is_differentiable_on 2, Z;
    f is_differentiable_on 1,Z by AS, TAYLOR_1:23; then
    A3: f is_differentiable_on Z & ((diff(f,Z)).1) = f`| Z by AS, DIFF1;
    (diff(f,Z)).2 = (diff(f,Z)).(1+1)
    .= (f`| Z)`| Z by TAYLOR_1:def 5, A3;
    hence thesis by AS, A3;
  end;
