reserve E, x, y, X for set;
reserve A, B, C, D for Subset of E^omega;
reserve a, a1, a2, b, c, c1, c2, d, ab, bc for Element of E^omega;
reserve e for Element of E;
reserve i, j, k, l, n, n1, n2, m for Nat;

theorem Th41:
  x in A* iff ex n st x in A |^ n
proof
  thus x in A* implies ex n st x in A |^ n
  proof
    defpred P[set] means ex k st $1 = A |^ k;
    assume x in A*;
    then consider X such that
A1: x in X and
A2: X in { B : ex k st B = A |^ k } 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 n such that
A4: x in A |^ n;
  defpred P[set] means ex k st $1 = A |^ k;
  consider B such that
A5: x in B and
A6: P[B] by A4;
  reconsider A = { C : P[C] } as Subset-Family of E^omega from DOMAIN_1:sch 7;
  B in A by A6;
  hence thesis by A5,TARSKI:def 4;
end;
