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 ( #Z n) *f is_differentiable_in x0 &
  diff( ( #Z n) *f,x0) = n * (f.x0) #Z (n-1) *diff(f,x0)
proof
  assume
A1: f is_differentiable_in x0;
A2: ( #Z n) is_differentiable_in f.x0 & x0 is Real by Th2;
  hence ( #Z n)*f is_differentiable_in x0 by A1,FDIFF_2:13;
  thus diff(( #Z n)*f,x0) = diff(( #Z n),f.x0)*diff(f,x0) by A1,A2,FDIFF_2:13
    .=n * (f.x0) #Z (n-1) *diff(f,x0) by Th2;
end;
