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 Th18:
  (A |^.. m) ^^ (A |^.. n) = A |^.. (m + n)
proof
  defpred P[Nat] means (A |^.. m) ^^ (A |^.. $1) = A |^.. (m + $1);
A1: now
    let n;
    assume
A2: P[n];
    (A |^.. m) ^^ (A |^.. (n + 1)) = (A |^.. m) ^^ ((A |^.. n) ^^ A) by Th16
      .= A |^.. (m + n) ^^ A by A2,FLANG_1:18
      .= A |^.. (m + n + 1) by Th16;
    hence P[n + 1];
  end;
  (A |^.. m) ^^ (A |^.. 0) = (A |^.. m) ^^ (A*) by Th11
    .= (A |^.. (m + 0)) by Th17;
  then
A3: P[0];
  for n holds P[n] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
