reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, a1, a2, b for Element of E^omega;
reserve i, k, l, m, n for Nat;

theorem Th19:
  n > 0 implies (A |^.. m) |^ n = A |^.. (m * n)
proof
  defpred P[Nat] means $1 > 0 implies (A |^.. m) |^ $1 = A |^.. (m * $1);
A1: now
    let n;
    assume
A2: P[n];
    now
      assume n + 1 > 0;
      per cases;
      suppose
        n = 0;
        hence (A |^.. m) |^ (n + 1) = A |^.. (m * (n + 1)) by FLANG_1:25;
      end;
      suppose
        n > 0;
        hence (A |^.. m) |^ (n + 1) = (A |^.. (m * n)) ^^ (A |^.. m) by A2,
FLANG_1:23
          .= A |^.. (m * n + m) by Th18
          .= A |^.. (m * (n + 1));
      end;
    end;
    hence P[n + 1];
  end;
A3: P[0];
  for n holds P[n] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
