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 Th2:
  m <= n & k <= l & m + k <= i & i <= n + l implies ex mn, kl st mn
  + kl = i & m <= mn & mn <= n & k <= kl & kl <= l
proof
  assume that
A1: m <= n and
A2: k <= l and
A3: m + k <= i and
A4: i <= n + l;
  per cases;
  suppose
    i <= n + k;
    then consider mn such that
A5: mn + k = i & m <= mn & mn <= n by A3,Th1;
    take mn, k;
    thus mn + k = i & m <= mn & mn <= n by A5;
    thus thesis by A2;
  end;
  suppose
    i > n + k;
    then consider kl such that
A6: kl + n = i and
A7: k <= kl & kl <= l by A4,Th1;
    take n, kl;
    thus n + kl = i & m <= n & n <= n by A1,A6;
    thus thesis by A7;
  end;
end;
