
theorem
for R being Ring
for p being Polynomial of R
for a,b being Element of R holds (a * b) * p = a * (b * p)
proof
let R be Ring, p be Polynomial of R; let a,b be Element of R;
now let o be object;
  assume o in NAT; then
  reconsider j = o as Element of NAT;
  thus ((a * b) * p).o
     = (a * b) * (p.j) by POLYNOM5:def 4
    .= a * (b * (p.j)) by GROUP_1:def 3
    .= a * ((b * p).j) by POLYNOM5:def 4
    .= (a * (b * p)).o by POLYNOM5:def 4;
  end;
hence thesis;
end;
