Multinilponent groups

Let $\mathfrak{N}_k$ be the variety of all nilpoteht groups of class $\leqslant k$. From the varieties $\mathfrak{N}_{k_1},\dots,\mathfrak{N}_{k_s}$ the variety $\mathfrak{N}$ is constructed by intersections and multiplications. Any group of variety $\mathfrak{N}$ is called the multipolynilpotent group. In this note is proved Malcev's hypothesis: free multipolynilpotent group $N$ satisfies the following conditions:

• $\bigcap\limits_n\gamma_n(N)=\{1\}$, where $\gamma_n(N)$ is $n$ member of descending central series of the group $N$, $n$ is natural number,
• factors $\gamma_n(N)/\gamma_{n+1}(N)$ are free abelian groups.