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 Th48:
  x in A+ iff ex n st n > 0 & x in A |^ n
proof
  thus x in A+ implies ex n st n > 0 & x in A |^ n
  proof
    defpred P[set] means ex n st n > 0 & $1 = A |^ n;
    assume x in A+;
    then consider X such that
A1: x in X and
A2: X in { B: ex n st n > 0 & B = A |^ n } 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: n > 0 and
A5: x in A |^ n;
  defpred P[set] means ex n st n > 0 & $1 = A |^ n;
  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;
