The structure of this talk is the following:

-The starting point is the straightening formula for bideterminants (Doubilet, Rota, Stein, 1974) and its analogs for Pfaffians and Gramians (De Concini, Procesi 1976).

- From the straightening formulae one derives versions of the E. Pascal theorems for scalar products, symplectic products and inner products. These results lead to the main combinatorial tool: the cancellation laws.

-The First Fundamental Theorem for vector invariants of G = GL(d); Sp2m; O(d) is proved into three steps (De Concini-Procesi, 1976): Consider the coordinate ring S of the affine variety on which the group G acts, and a subalgebra B of S^G which we want to show equals S^G.

Consider the localization with respect to a suitable Zariski open set and prove the equality between the localized rings S[1/\Delta]^G = B[1/\Delta]; \Delta\in B: Let \psi\in S^G \subset S[1/\Delta]^G = B[1/\Delta]. There exists an integer k such that \Delta^k \psi\in B; the cancellation laws imply that \psi\in B.

Therefore, the First Fundamental Theorems follows: S^G = B.