reserve E, F, G,S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th50:
  for i be Nat,
      E,F be RealNormSpace,
      Z be non empty Subset of E,
      L1 be PartFunc of E,F,
      L0 be Point of F
   st Z is open & L1 = Z --> L0
  holds
    L1 is_differentiable_on i,Z
  & diff(L1,i,Z)`|Z is_continuous_on Z
proof
  let n be Nat,
      E,F be RealNormSpace,
      Z be non empty Subset of E,
      L1 be PartFunc of E,F,
      L0 be Point of F;
  assume A1: Z is open & L1 = Z --> L0;
  then A2: dom L1 = Z by FUNCOP_1:13;

  for i be Nat st i <= n - 1
  holds diff(L1,i,Z) is_differentiable_on Z by A1,Th49;
  hence L1 is_differentiable_on n,Z by A2,NDIFF_6:14;
  thus diff(L1,n,Z)`|Z is_continuous_on Z by A1,Th49;
end;
