reserve n for Nat,
  i for Integer,
  p, x, x0, y for Real,
  q for Rational,
  f for PartFunc of REAL,REAL;

theorem Th17:
  exp_R" is_differentiable_on dom (exp_R") & for x be Real
  st x in dom (exp_R") holds diff(((exp_R)"),x) = 1/x
proof
  thus exp_R" is_differentiable_on dom (exp_R") by Th16,FDIFF_2:45;
  let x be Real such that
A1: x in dom ((exp_R)");
A2: x in rng (exp_R) by A1,FUNCT_1:33;
  diff(exp_R, ((exp_R)").x)=exp_R.( ((exp_R)").x) by Th16
    .=x by A2,FUNCT_1:35;
  hence thesis by A1,Th16,FDIFF_2:45;
end;
