Hilbert–Mal’tsev varieties

The author's paper (“On varieties of associative algebras with local finiteness conditions”, Algebra i Analiz 9 (1997), no. 4, 119–174) gives a complete description of Hilbert–Mal'tsev varieties (these are the ones  in which classical finiteness conditions are locally satisfied).  An answer to a number of well-known questions in this subject was obtained as consequences. Unfortunately, in the mentioned paper, in two important auxiliary statements there are math inexactitude. This gave reason to doubt the correctness of some of the results obtained. In this paper, corrected proofs of these statements are given.