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

theorem Th95:
  (A?) |^ k = A |^ (0, k)
proof
  defpred P[Nat] means (A?) |^ $1 = A |^ (0, $1);
A1: now
    let k;
    assume
A2: P[k];
    (A?) |^ (k + 1) = ((A?) |^ k) ^^ ((A?) |^ 1) by FLANG_1:33
      .= A |^ (0, k) ^^ (A?) by A2,FLANG_1:25
      .= A |^ (0, k) ^^ (A |^ (0, 1)) by Th79
      .= A |^ (0 + 0, k + 1) by Th37;
    hence P[k + 1];
  end;
  (A?) |^ 0 = {<%>E} by FLANG_1:24
    .= A |^ (0, 0) by Th45;
  then
A3: P[0];
  for k holds P[k] from NAT_1:sch 2(A3, A1);
  hence thesis;
end;
