Corollary 1

Let $(e_1, . . . , e_n)$ be a basis of V and $t$ an element of $D^*_{ab}$. There exists a unique element $\omega$ of Ω(V) such that $\omega (e_1, . . . , e_n) =t$.

Let us prove the corollary by induction on $n$. If $n= 0$, then the only basis of V is the empty one, so the assertion is true. Now suppose $n\geqslant 1$; denote by W the subspace of V generated by $e_1, . . . , e_{n-1}$. By Proposition 1, there exists a bijection Λ from Ω(V) to Ω(W) satisfying

for every basis $(w_1, . . . , w_{n-1})$ of W and every $\omega$ in Ω(V). By the induction hypothesis, there exists a unique element $\varphi$ of Ω(W) such that $\varphi (e_1, . . . , e_{n-1}) =t$. The relation $\omega (e_1, . . . , e_n) =t$, for $\omega$ in Ω(V), is equivalent to $\Lambda (\omega ) =\varphi$. Corollary 1 follows.