Corollary 1

Let, for i = 1, 2, E_i be a finite-dimensional vector space, Φ_i a non-degenerate ε-hermitian form on E_i satisfying (T) (resp. Q_i a non-degenerate quadratic form on E_i), E'_i and E''_i two orthogonal subspaces of E_i whose direct sum is E_i. If the forms Φ_1 and Φ_2 (resp. Q_1 and Q_2) are equivalent, and if their restrictions to E'_1 and E'_2 are equivalent, the same is true of their restrictions to E''_1 and E''_2.

Indeed, let u be a metric isomorphism of E'_1 onto E'_2. By Theorem 1, u extends to a metric isomorphism ϕ of E_1 onto E_2. Since Φ_i is non-degenerate, E''_i is the orthogonal of E'_i in E_i, hence ϕ maps E''_1 onto E''_2. QED.