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 Th51:
  for n be Nat,
      S be RealNormSpace,
      Z be Subset of S,
      f be PartFunc of S,S
   st Z is open & f = id [#]S
  holds
    f is_differentiable_on n,Z
  & diff(f,n,Z) is_continuous_on Z
proof
  let n be Nat,
      S be RealNormSpace,
      Z be Subset of S,
      f be PartFunc of S,S;
  assume A1: Z is open & f = id [#]S;

  reconsider L = id [#]S as Lipschitzian LinearOperator of S,S by LOPBAN_2:3;

  A2: dom L = [#]S;
  for i be Nat st i <= n - 1
  holds diff(L,i,[#]S) is_differentiable_on [#]S by NDIFF12:20;
  then A3: f is_differentiable_on n,[#]S by A1,A2,NDIFF_6:14;
  diff(L,n,[#]S) is_continuous_on [#]S by NDIFF_1:45,NDIFF12:20;
  hence thesis by A1,A3,Th10;
end;
