
theorem Th27:
  for L being comRing, i, j being Nat, p being
  Polynomial of L holds (p`^i) *' (p`^j) = p `^(i+j)
proof
  let L be comRing, i, j being Nat, p be Polynomial of L;
  defpred P[Nat] means (p`^i) *' (p`^$1) = p `^(i+$1);
A1: for j being Nat st P[j] holds P[j+1]
  proof
    let j be Nat such that
A2: P[j];
    (p`^i) *' (p`^(j+1)) = (p`^i) *' ((p`^j) *' p) by POLYNOM5:19
      .= (p`^(i+j)) *' p by A2,POLYNOM3:33
      .= p`^(i+j+1) by POLYNOM5:19
      .= p`^(i+(j+1));
    hence thesis;
  end;
  (p`^i) *' (p`^0) = (p`^i) *' 1_. L by POLYNOM5:15
    .= (p`^(i+(0 qua Nat))) by POLYNOM3:35;
  then
A3: P[ 0 ];
  for j being Nat holds P[j] from NAT_1:sch 2(A3,A1);
  hence thesis;
end;
