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 Th10:
  for S,T be RealNormSpace,
      f be PartFunc of S,T,
      X,Z be Subset of S
   st Z is open & Z c= X
  holds
    for i be Nat
     st f is_differentiable_on i,X
      & diff(f,i,X) is_continuous_on X
    holds
      f is_differentiable_on i,Z
    & diff(f,i,Z) is_continuous_on Z
proof
  let S,T be RealNormSpace,
      f be PartFunc of S,T,
      X,Z be Subset of S;
  assume A1: Z is open & Z c= X;
  let i be Nat;
  assume
  A2: f is_differentiable_on i,X
    & diff(f,i,X) is_continuous_on X;
  then diff(f,i,X) is_continuous_on Z by A1,NFCONT_1:23;
  then diff(f,i,X) | Z is_continuous_on Z by NFCONT_1:21;
  hence thesis by A1,A2,Th9;
end;
