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 Th3:
  for S,T be RealNormSpace,
      f be PartFunc of S,T,
      Z be Subset of S
  holds
    f|Z is_differentiable_on Z
      iff
    f is_differentiable_on Z
proof
  let S,T be RealNormSpace,
      f be PartFunc of S,T,
      Z be Subset of S;

  hereby
    assume
    A1: f|Z is_differentiable_on Z;
    dom(f|Z) c= dom f by RELAT_1:60;
    hence f is_differentiable_on Z by A1,XBOOLE_1:1;
  end;

  thus f is_differentiable_on Z
    implies f|Z is_differentiable_on Z by RELAT_1:62;
end;
