reserve Z for open Subset of REAL;

theorem Th6:
  for n be Nat holds diff(exp_R,Z).n=exp_R | Z
proof
  let n be Nat;
  defpred P[Nat] means diff(exp_R,Z).$1 = exp_R | Z;
A1: for k be Nat st P[k] holds P[k+1]
  proof
    let k be Nat such that
A2: P[k];
A3: exp_R is_differentiable_on Z by FDIFF_1:26,TAYLOR_1:16;
    diff(exp_R,Z).(k+1)=(diff(exp_R,Z).k) `| Z by TAYLOR_1:def 5
      .=exp_R `| Z by A2,A3,FDIFF_2:16
      .=exp_R | Z by Th5;
    hence thesis;
  end;
A4: P[0] by TAYLOR_1:def 5;
  for n be Nat holds P[n] from NAT_1:sch 2(A4,A1);
  hence thesis;
end;
