Lemma 3

Let B be a Hall set relative to I and let $H_{r,s} = \sum_{b \in B} \lambda_b e_b$ be the decomposition of $H_{r,s}$ with respect to the corresponding basis of $L(\{U, V\})$. Then

This is trivial for $p = 0$, for $\lambda_b \in \mathbf{Q}$; and it follows from Proposition 1 of no. 1 for $p \neq 0$.

Moreover,

We show more generally by induction on $n$ that, for every alternant $b$ of degree $n$ in the two indeterminates U and V (§ 2, no. 6), $\| \tilde{b} \| \leq 1$. If $n = 1$, $\tilde{b}$ is one of the projections of $g \times g$ onto $g$ and hence is of norm $\leq 1$; if $n > 1$, there exist two alternants $b_1$ and $b_2$ of degrees $< n$ such that $b = [b_1, b_2]$. As the mapping $\gamma : (x, y) \mapsto [x, y]$ of $g \times g$ into $g$ is bilinear of norm $\leq 1$, we have (Differentiable and Analytic Manifolds, R, Appendix, no. 4) Relations (11) and (12) imply the lemma.