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 Th38:
  A |^ (m + 1, n + 1) = (A |^ (m, n)) ^^ A
proof
  per cases;
  suppose
    m <= n;
    hence A |^ (m + 1, n + 1) = (A |^ (m, n)) ^^ (A |^ (1, 1)) by Th37
      .= (A |^ (m, n)) ^^ (A |^ 1) by Th22
      .= (A |^ (m, n)) ^^ A by FLANG_1:25;
  end;
  suppose
A1: m > n;
    then A |^ (m, n) = {} by Th21;
    then
A2: (A |^ (m, n)) ^^ A = {} by FLANG_1:12;
    m + 1 > n + 1 by A1,XREAL_1:8;
    hence thesis by A2,Th21;
  end;
end;
