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 Th33:
  for E,F be RealNormSpace,
      n be Nat,
      Z be Subset of E,
      g be PartFunc of E,F
   st g`|Z is_differentiable_on n,Z
      & g is_differentiable_on Z
      & diff(g`|Z,n,Z) is_continuous_on Z
  holds g is_differentiable_on n+1,Z
     & diff(g,n+1,Z) is_continuous_on Z
proof
  let E,F be RealNormSpace,
      n be Nat,
      Z be Subset of E,
      g be PartFunc of E,F;
  assume
  A1: g`|Z is_differentiable_on n,Z
    & g is_differentiable_on Z
    & diff(g`|Z,n,Z) is_continuous_on Z;
  hence g is_differentiable_on n+1,Z by Th32;

  dom(g`|Z) = Z by A1,NDIFF_1:def 9;
  then A2: (g`|Z) | Z = (g|Z) `| Z by A1,Th4;
  diff(g,n+1,Z) = diff(g`|Z,n,Z) by A2,Th31;
  hence diff(g,n+1,Z) is_continuous_on Z by A1,Th30;
end;
