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,
      X,Z be Subset of S
  st Z is open & Z c= X
   & f is_differentiable_on X
   & f `| X is_continuous_on X
  holds f `| 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
    & f is_differentiable_on X
    & f `| X is_continuous_on X;
  then A2: f `| Z = (f`| X) | Z by Th5;
  f `| X is_continuous_on Z by A1,NFCONT_1:23;
  hence thesis by A2,NFCONT_1:21;
end;
