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 Th23:
  A ^^ (A |^.. n) = (A |^.. n) ^^ A
proof
  defpred P[Nat] means A ^^ (A |^.. $1) = (A |^.. $1) ^^ A;
A1: now
    let k be Nat;
    assume
A2: P[k];
    A ^^ (A |^.. (k + 1)) = A ^^ ((A |^.. k) ^^ A) by Th16
      .= (A |^.. k) ^^ A ^^ A by A2,FLANG_1:18
      .= (A |^.. (k + 1)) ^^ A by Th16;
    hence P[k + 1];
  end;
  A ^^ (A |^.. 0) = A ^^ (A*) by Th11
    .= (A*) ^^ A by FLANG_1:57
    .= (A |^.. 0) ^^ A by Th11;
  then
A3: P[0];
  for k being Nat holds P[k] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
