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 Th37:
  A c= B implies A |^ n c= B |^ n
proof
  defpred P[Nat] means A |^ $1 c= B |^ $1;
  assume
A1: A c= B;
A2: now
    let n;
    assume
A3: P[n];
    (A |^ n) ^^ A = A |^ (n + 1) & (B |^ n) ^^ B = B |^ (n + 1) by Th23;
    hence P[n + 1] by A1,A3,Th17;
  end;
  A |^ 0 = {<%>E} by Th24;
  then
A4: P[0] by Th24;
  for n holds P[n] from NAT_1:sch 2(A4, A2);
  hence thesis;
end;
