
theorem der3:
for R being Ring
for p being Element of the carrier of Polynom-Ring R
for a being Element of R
holds (Deriv R).(a*p) = a * (Deriv R).p
proof
let F be Ring, p be Element of the carrier of Polynom-Ring F;
let a be Element of F;
reconsider r = a*p as Element of the carrier of Polynom-Ring F
   by POLYNOM3:def 10;
now let o be object;
  assume o in NAT; then
  reconsider i = o as Element of NAT;
  ((Deriv F).r).i
     = (i+1) * (r.(i+1)) by RINGDER1:def 8
    .= (i+1) * (a * p.(i+1)) by POLYNOM5:def 4
    .= ((i+1) * a) * p.(i+1) by BINOM:19
    .= (a * (i+1)) * p.(i+1) by BINOM:17
    .= a * ((i+1) * p.(i+1)) by BINOM:21
    .= a * ((Deriv F).p).i by RINGDER1:def 8
    .= (a * (Deriv F).p).i by POLYNOM5:def 4;
  hence ((Deriv F).r).o = (a * (Deriv F).p).o;
  end;
hence thesis by FUNCT_2:12;
end;
