Remark

Let $(h_1, ..., h_r)$ be a finite generating system of the vector space $H_1$ over $H_0$. For $\lambda = (\lambda_1, ..., \lambda_r) \in H_0^r$, put $h_\lambda = \lambda_1 h_1 + \cdots + \lambda_r h_r$. The proofs of Prop. 6 and Corollary 1 entail the following result: the set of elements $(\lambda_1, ..., \lambda_n)$ of $(H_0^r)^n$ such that the elements $a_i = h_{\lambda_i} \in H_1$ satisfy the conclusion of Corollary 1 contains the complement of the union of a finite number of vector subspaces of $(H_0^r)^n$ distinct from the whole space.