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
  for S, T be RealNormSpace,
      f be PartFunc of S,T,
      Z be Subset of S
  holds
    for k be Nat
    st f is_differentiable_on k,Z
     & diff(f,k,Z) is_continuous_on Z
    holds
      f|Z is_differentiable_on k,Z
    & diff(f|Z,k,Z) is_continuous_on Z by Th7;
