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 Th39:
  (A |^ (m, n)) ^^ (A |^ (k, l)) = (A |^ (k, l)) ^^ (A |^ (m, n))
proof
  per cases;
  suppose
A1: m <= n & k <= l;
    hence (A |^ (m, n)) ^^ (A |^ (k, l)) = A |^ (m + k, n + l) by Th37
      .= (A |^ (k, l)) ^^ (A |^ (m, n)) by A1,Th37;
  end;
  suppose
    m > n;
    then
A2: A |^ (m, n) = {} by Th21;
    then (A |^ (m, n)) ^^ (A |^ (k, l)) = {} by FLANG_1:12;
    hence thesis by A2,FLANG_1:12;
  end;
  suppose
    k > l;
    then
A3: A |^ (k, l) = {} by Th21;
    then (A |^ (m, n)) ^^ (A |^ (k, l)) = {} by FLANG_1:12;
    hence thesis by A3,FLANG_1:12;
  end;
end;
