
theorem BBD:
for R being Ring
for p being Polynomial of R
for a,b being Element of R holds (a + b) * p = a * p + 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 * (p.j) + b * (p.j) by VECTSP_1:def 3
    .= (a * p).j + b * (p.j) by POLYNOM5:def 4
    .= (a * p).j + (b * p).j by POLYNOM5:def 4
    .= (a * p + b * p).o by NORMSP_1:def 2;
  end;
hence thesis;
end;
