reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem
  for G being finite Group,a,b being Element of G st
  G is p-commutative-group-like
  holds for n holds (a * b) |^ (p |^n) = a |^ (p |^n) * (b |^ (p |^n))
proof
  let G be finite Group;
  let a,b be Element of G;
  assume
A1: G is p-commutative-group-like;
  defpred P[Nat] means for n holds
  (a * b) |^ (p |^ $1) = a |^ (p |^$1) * (b |^ (p |^$1));
A2: (a * b) |^ (p |^0) = (a * b) |^ 1 by NEWTON:4
                      .= a * b by GROUP_1:26;
  a |^ (p |^0) * (b |^ (p |^0)) = a |^ 1 * (b |^ (p |^0)) by NEWTON:4
                               .= a * (b |^ (p |^0)) by GROUP_1:26
                               .= a * (b |^1) by NEWTON:4
                               .= a * b by GROUP_1:26;
  then
A3: P[0] by A2;
A4: now
    let n;
    assume
A5: P[n];
    set a1 = a |^ (p |^n), b1 = b |^ (p |^n);
 (a * b) |^ (p |^ (n + 1)) = (a * b) |^ (p |^ n * p) by NEWTON:6
                .= ((a * b) |^ (p |^ n)) |^ p by GROUP_1:35
                .= (a |^ (p |^n) * (b |^ (p |^n))) |^ p by A5
                .= a1 |^ p * (b1 |^ p) by A1
                .= a |^ ((p |^n) * p) * (b |^ (p |^n) |^ p) by GROUP_1:35
                .= a |^ ((p |^n) * p) * (b |^ ((p |^n) * p)) by GROUP_1:35
                .= a |^ (p |^(n +1)) * (b |^ ((p |^n) * p)) by NEWTON:6
                .= a |^ (p |^(n +1)) * (b |^ (p |^(n +1))) by NEWTON:6;
    hence P[n+1];
    end;
  for n holds P[n] from NAT_1:sch 2(A3,A4);
  hence thesis;
end;
