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 Th14:
f is_differentiable_on n,Z
  iff
Z c= dom f & for i be Nat st i <= n-1 holds diff(f,i,Z) is_differentiable_on Z
proof
   hereby assume A1: f is_differentiable_on n,Z;
    hence Z c= dom f;
    hereby let i be Nat;
     assume i <= n-1; then
     modetrans(diff(f,Z).i,S,diff_SP(i,S,T))
       is_differentiable_on Z by A1;
     hence diff(f,i,Z) is_differentiable_on Z by Def4;
    end;
   end;
   assume A2: Z c= dom f
           & for i be Nat st i<=n-1 holds diff(f,i,Z) is_differentiable_on Z;
   now let i be Nat;
    assume i <= n-1; then
    diff(f,i,Z) is_differentiable_on Z by A2;
    hence modetrans(diff(f,Z).i,S,diff_SP(i,S,T))
      is_differentiable_on Z by Def4;
   end;
   hence f is_differentiable_on n,Z by A2;
end;
