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 DIFF1:
  for f be PartFunc of REAL,REAL, Z be Subset of REAL
  st Z c= dom f & Z is open & f is_differentiable_on 1,Z holds
  f is_differentiable_on Z & ((diff(f,Z)).1) = f`| 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 1, Z; then
    (diff(f,Z)).0 is_differentiable_on Z;
    then (f|Z) is_differentiable_on Z by TAYLOR_1:def 5;
    hence
    X1: f is_differentiable_on Z by LM003, AS;
    thus (diff(f,Z)).1 = (diff(f,Z)).(0+1)
    .= ((diff(f,Z)).0) `| Z by TAYLOR_1:def 5
    .= ((f|Z)) `| Z by TAYLOR_1:def 5
    .= f `| Z by AS, X1, LM00;
  end;
