reserve Z for open Subset of REAL;

theorem Th10:
  for n be Element of NAT holds exp_R is_differentiable_on n, Z
proof
  let n be Element of NAT;
  let i be Nat such that
  i <= n-1;
  reconsider i as Element of NAT by ORDINAL1:def 12;
A1: for x be Real
   st x in Z holds diff(exp_R,Z).i | Z is_differentiable_in x
  proof
A2: diff(exp_R,Z).i | Z = exp_R | Z | Z by Th6
      .= exp_R | Z by FUNCT_1:51;
A3: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
    let x be Real;
    assume x in Z;
    hence thesis by A2,A3,FDIFF_1:def 6;
  end;
  dom(diff(exp_R,Z).i) = dom(exp_R | Z) by Th6
    .= Z by Th5;
  hence thesis by A1,FDIFF_1:def 6;
end;
