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
  A* ^^ (A |^ k) = A |^.. k
proof
  defpred P[Nat] means A* ^^ (A |^ $1) = A |^.. $1;
A1: now
    let k;
    assume
A2: P[k];
    A* ^^ (A |^ (k + 1)) = A* ^^ ((A |^ k) ^^ A) by FLANG_1:23
      .= A |^.. k ^^ A by A2,FLANG_1:18
      .= A |^.. (k + 1) by Th16;
    hence P[k + 1];
  end;
  A* ^^ (A |^ 0) = A* ^^ {<%>E} by FLANG_1:24
    .= A* by FLANG_1:13
    .= A |^.. 0 by Th11;
  then
A3: P[0];
  for k holds P[k] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
