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 Th23:
  A |^ (n + 1) = (A |^ n) ^^ A
proof
  consider concat being sequence of  bool (E^omega) such that
A1: A |^ n = concat.n and
A2: concat.0 = {<%>E} and
A3: for i holds concat.(i + 1) = concat.i ^^ A by Def2;
  concat.(n + 1) = (A |^ n) ^^ A by A1,A3;
  hence thesis by A2,A3,Def2;
end;
