Corollary 1

The family $(\phi(x))_{x \in X}$ is free over K in $L(X)$.

Let $x_1, x_2, \ldots, x_n$ be distinct elements of X and $\lambda_1, \ldots, \lambda_n$ be elements of K such that

Let $g$ be the commutative Lie algebra with K as underlying module. For $i = 1, 2, \ldots, n$, there exists a homomorphism $F_i$ of $L(X)$ into $g$ such that $F_i(\phi(x_i)) = 1$ and $F_i(\phi(x)) = 0$ for $x \neq x_i$ (Proposition 1); applying $F_i$ to relation (3), we obtain $\lambda_i = 0$.