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

theorem
  f is_differentiable_in x0 implies exp_R*f is_differentiable_in x0 &
  diff(exp_R*f,x0) = exp_R.(f.x0)*diff(f,x0)
proof
  assume
A1: f is_differentiable_in x0;
A2: x0 is Real & exp_R is_differentiable_in f.x0 by Th16,FDIFF_1:9
,XREAL_0:def 1;
  hence exp_R*f is_differentiable_in x0 by A1,FDIFF_2:13;
  thus diff(exp_R*f,x0) = diff(exp_R,f.x0)*diff(f,x0) by A1,A2,FDIFF_2:13
    .=exp_R.(f.x0) *diff(f,x0) by Th16;
end;
