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
  (A |^ (m, n)) \/ (B |^ (m, n)) c= (A \/ B) |^ (m, n)
proof
  let x be object;
  assume
A1: x in (A |^ (m, n)) \/ (B |^ (m, n));
  per cases by A1,XBOOLE_0:def 3;
  suppose
    x in A |^ (m, n);
    then consider mn such that
A2: m <= mn & mn <= n and
A3: x in A |^ mn by Th19;
A4: (A |^ mn) \/ (B |^ mn) c= (A \/ B) |^ mn by FLANG_1:38;
    x in (A |^ mn) \/ (B |^ mn) by A3,XBOOLE_0:def 3;
    hence thesis by A2,A4,Th19;
  end;
  suppose
    x in B |^ (m, n);
    then consider mn such that
A5: m <= mn & mn <= n and
A6: x in B |^ mn by Th19;
A7: (A |^ mn) \/ (B |^ mn) c= (A \/ B) |^ mn by FLANG_1:38;
    x in (A |^ mn) \/ (B |^ mn) by A6,XBOOLE_0:def 3;
    hence thesis by A5,A7,Th19;
  end;
end;
