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
  m <= n implies A |^ (m, n) = A |^ m \/ A |^ (m + 1, n)
proof
  assume
A1: m <= n;
  per cases by A1,XXREAL_0:1;
  suppose
    m < n;
    then A |^ (m, n) = A |^ (m, m) \/ A |^ (m + 1, n) by Th25;
    hence thesis by Th22;
  end;
  suppose
A2: m = n;
    then
A3: m + 1 > n by NAT_1:13;
    thus A |^ (m, n) = A |^ m \/ {} by A2,Th22
      .= A |^ m \/ A |^ (m + 1, n) by A3,Th21;
  end;
end;
