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 Th2:
  x in A |^.. n iff ex m st n <= m & x in A |^ m
proof
  thus x in A |^.. n implies ex m st n <= m & x in A |^ m
  proof
    defpred P[set] means ex m st n <= m & $1 = A |^ m;
    assume x in A |^.. n;
    then consider X such that
A1: x in X and
A2: X in { B: ex m st n <= m & B = A |^ m } by TARSKI:def 4;
A3: X in { B: P[B] } by A2;
    P[X] from CARD_FIL:sch 1(A3);
    hence thesis by A1;
  end;
  given m such that
A4: n <= m and
A5: x in A |^ m;
  defpred P[set] means ex m st n <= m & $1 = A |^ m;
  consider B such that
A6: x in B and
A7: P[B] by A4,A5;
  reconsider A = { C : P[C] } as Subset-Family of E^omega from DOMAIN_1:sch 7;
  B in A by A7;
  hence thesis by A6,TARSKI:def 4;
end;
