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
  (A*) ^^ (A*) = A*
proof
  <%>E in A* by Th48;
  then
A1: A* c= (A*) ^^ (A*) by Th16;
  (A*) ^^ (A*) c= A* by Th46;
  hence thesis by A1,XBOOLE_0:def 10;
end;
