The algebras M-a,M-b(E) ⊗ E and Ma+b(E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984-1987 (see Kemer, 1991); other proofs of it were given by Regev (1990), and in several particular cases, by Di Vincenzo (1992), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M-1,M-1 (E) ⊗ E and M-2 (E) when the base field is infinite and of characteristic p > 2. The algebra M-a,M-a(E) ⊗ E satisfies certain graded identities that are not satisfied by M-2a (E). In another paper we proved that the algebras M-1,M-1 (E) and E ⊗ E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.33410111022