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 & f.x0 > 0 implies ( #R p) *f
is_differentiable_in x0 & diff( ( #R p) *f,x0) = p*( (f.x0) #R (p-1)) * diff(f,
  x0)
proof
  assume that
A1: f is_differentiable_in x0 and
A2: f.x0 > 0;
A3: ( #R p) is_differentiable_in f.x0 by A2,Th21;
  hence ( #R p) *f is_differentiable_in x0 by A1,FDIFF_2:13;
  thus diff( ( #R p) *f,x0) = diff( ( #R p),f.x0)*diff(f,x0) by A1,A3,
FDIFF_2:13
    .=p*( (f.x0) #R (p-1))*diff(f,x0) by A2,Th21;
end;
