Оn direct decompositions of $n$-multiply $\omega$-saturated formations

All groups considered are finite. Let $\{\mathfrak{F}_i \mid i\in I\}$ be a set of non-empty subclasses of a class of groups $\mathfrak{F}$ such that $\mathfrak{F}_i \cap \mathfrak{F}_j = (1)$ for all distinct $i, j \in I$. We write $\mathfrak{F}=\bigoplus_{i\in I}\mathfrak{F}_i$ to denote the collection of all groups of the form $А_1\times \dots \times А_t$, where $A_1 \in \mathfrak{F}_{i_1},\dots,A_t \in \mathfrak{F}_{i_1}$ for some $i_1,\dots, i_t \in I$. We proved the following theorem.
Theorem. Let $\mathfrak{F}=\bigoplus_{i \in I} \mathfrak{F}_i$ where $\mathfrak{F}_i$ is a formation. Then $\mathfrak{F}$ is $n$-multiply ($n\ge 1$)$\omega$-saturated formation if and only if $\mathfrak{F}_i$ is $n$-multiply $\omega$-saturated for all $i \in I$.