Corollary 2

Let f be an A-linear mapping of E into a vector space F over K; if Ker(f) ⊂ T(E), the K-linear mapping $\vec{f}$ associated with f is injective.

We write an element of Ker($\vec{f}$) in the form $\lambda^{-1}\phi(x)$, where $\lambda \in A, \lambda \neq 0, x \in E$; the relation $\vec{f}(\lambda^{-1}\phi(x)) = 0$ is equivalent to $\lambda^{-1}\vec{f}(\phi(x)) = 0$ in F and hence to $f(x) = \vec{f}(\phi(x)) = 0$. By hypothesis, this implies $x \in T(E)$ and hence $\phi(x) = 0$, which proves the corollary.