The present and the next three chapters state and prove 20 lemmas used in the KAM proof of the foregoing chapter. Lemmas G1, G5 and D3 are important theorems in themselves (G1 is equivalent to an important lemma by Brouwer 1911a). The following four ‘analytic lemmas’ numbered A1 to A4 are contained in section 4.2 of Arnold (1963a) and give limits for the Fourier coefficients and the derivatives of analytic functions.