One proof of the original Kemer's theorems (concerning the text of C. Procesi "What happened to PI-theory", arxiv.org/abs/1403.5673)

We consider associative algebras over a field of characteristic zero. We give a version of the proof of the Kemer's theorems concerning the Specht problem solution. It is proved that the ideal of graded identities of a finitely generated PI-superalgebra coincides with the ideal of graded identities of some finite dimensional superalgebra. This implies that the ideal of polynomial identities of any (not necessary finitely generated) PI-algebra coincides with the ideal of identities of the Grassmann envelope of a finite dimensional superalgebra, and is finitely generated as a T-ideal.