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 Th73:
  x in A? iff ex k st k <= 1 & x in A |^ k
proof
  thus x in A? implies ex k st k <= 1 & x in A |^ k
  proof
    defpred P[set] means ex k st k <= 1 & $1 = A |^ k;
    assume x in A?;
    then consider X such that
A1: x in X and
A2: X in { B: ex k st k <= 1 & 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 k such that
A4: k <= 1 & x in A |^ k;
  defpred P[set] means ex k st k <= 1 & $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;
