Note that Lemma A is sufficient to prove that "e" is irrational, since otherwise we may write "e" = "p" / "q", where both "p" and "q" are non-zero integers, but by Lemma A we would have "qe" − "p" ≠ 0, which is a contradiction. Lemma A also suffices to prove that is irrational, since otherwise we may write = "k" / "n", where both "k" and "n" are integers) and then ±"i" are the roots of "n"2"x"2 + "k"2 = 0; thus 2 − 1 − 1 = 2"e"0 + "e""i" + "e"−"i" ≠ 0; but this is false.