reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th18:
for n be Nat, f be PartFunc of S,T
  st 1 <= n & f is_differentiable_on n,Z holds Z is open
proof
   let n be Nat, f be PartFunc of S,T;
   assume 1 <= n & f is_differentiable_on n,Z; then
   f is_differentiable_on 1,Z by Th17; then
   Z c= dom f & f|Z is_differentiable_on Z by Th15;
   hence Z is open by NDIFF_1:32;
end;
