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 Th17:
for S,T be RealNormSpace, f be PartFunc of S,T,
    Z be Subset of S, n be Nat st
 f is_differentiable_on n,Z for m be Nat st m <= n
  holds f is_differentiable_on m, Z
proof
   let S,T be RealNormSpace,
       f be PartFunc of S,T, Z be Subset of S;
   let n be Nat such that
A1: f is_differentiable_on n,Z;
   let m be Nat;
   assume m <=n; then
A2:m-1 <=n-1 by XREAL_1:13;
   thus Z c= dom f by A1;
   thus thesis by A1,A2,XXREAL_0:2;
end;
