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 Th19:
for n being Nat, f being PartFunc of S,T st 1<=n & f is_differentiable_on n,Z
 holds
  for i being Nat st i <= n holds
    diff_SP(S,T).i is RealNormSpace
  & diff(f,Z).i is PartFunc of S,diff_SP(i,S,T) & dom diff(f,i,Z) = Z
proof
   let n be Nat, f be PartFunc of S,T;
   assume A1: 1<=n & f is_differentiable_on n,Z;
   let i be Nat;
   assume A2: i<=n;
   diff_SP(i,S,T) is RealNormSpace;
   hence diff_SP(S,T).i is RealNormSpace;
   diff(f,i,Z) is PartFunc of S,diff_SP(i,S,T);
   hence diff(f,Z).i is PartFunc of S,diff_SP(i,S,T);
   per cases;
   suppose i = 0; then
    f|Z = diff(f,Z).i by Def5;
    hence dom diff(f,i,Z) = Z by RELAT_1:62,A1;
   end;
   suppose i <> 0; then
    reconsider i1=i-1 as Element of NAT by INT_1:3;
A3: diff(f,i1+1,Z) = diff(f,i1,Z)`|Z by Th13;
    diff(f,i1,Z) is_differentiable_on Z by A1,XREAL_1:9,A2,Th14;
    hence dom diff(f,i,Z) = Z by A3,NDIFF_1:def 9;
   end;
end;
