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 Th30:
  <%>E in A implies <%>E in A |^ n
proof
  defpred P[Nat] means <%>E in A |^ $1;
  assume
A1: <%>E in A;
A2: now
    let n;
    assume P[n];
    then <%>E in (A |^ n) ^^ A by A1,Th15;
    hence P[n + 1] by Th23;
  end;
  A |^ 0 = {<%>E} by Th24;
  then
A3: P[0] by TARSKI:def 1;
  for n holds P[n] from NAT_1:sch 2(A3, A2);
  hence thesis;
end;
