Corollary

— Let $P$ be a subfield of $K$. The canonical mapping $\alpha$ of $\Omega_P(K) \otimes_K L$ into $\Omega_P(L)$ is injective and the family $(d_{L/P} x_i)_{i \in I}$, is a basis (over $L$) of a subspace of $\Omega_P(L)$ supplementary to the image of $\alpha : \Omega_P(K) \otimes_K L \to \Omega_P(L)$.

Prop. 3, a) shows that every $P$-derivation of $K$ into a vector space $V$ over $L$ extends to a $P$-derivation of $L$ into $V$; the injectivity of $\alpha$ then follows by Prop. 2. The second assertion of the corollary follows from the exactness of the sequence $(\mathbf{E}_{P,K,L})$ and Prop. 3, b) (taking Formula (4) into account).