reserve E, x, y, X for set;
reserve A, B, C for Subset of E^omega;
reserve a, b for Element of E^omega;
reserve i, k, l, kl, m, n, mn for Nat;

theorem Th4:
  a ^ b = a or b ^ a = a implies b = {}
proof
  assume
A1: a ^ b = a or b ^ a = a;
  per cases by A1;
  suppose
A2: a ^ b = a;
    len(a ^ b) = len(a) + len(b) by AFINSQ_1:17;
    hence thesis by A2;
  end;
  suppose
A3: b ^ a = a;
    len(b ^ a) = len(b) + len(a) by AFINSQ_1:17;
    hence thesis by A3;
  end;
end;
