Order statistics-equivalence is finite-direct product-closed

Statement for direct product of two groups
Suppose $$G_1, G_2, H_1, H_2$$ are finite groups such that $$G_1$$ is order statistics-equivalent to $$H_1$$ and $$G_2$$ is order statistics-equivalent to $$H_2$$. Then, the fact about::external direct product $$G_1 \times G_2$$ is order statistics-equivalent to the external direct product $$H_1 \times H_2$$.

Here, two finite groups are termed order statistics-equivalent finite groups if they have the same order statistics, i.e., the same number of elements of each order.

Statement for direct product of finitely many groups
Suppose $$G_1, G_2, \dots, G_n$$ and $$H_1, H_2, \dots, H_n$$ are groups such that, for each $$i$$ satisfying $$1 \le i \le n$$, $$G_i$$ is order statistics-equivalent to $$H_i$$. Then, the fact about::external direct product $$G_1 \times G_2 \times \dots \times G_n$$ is order statistics-equivalent to $$H_1 \times H_2 \times \dots \times H_n$$.

Related facts

 * 1-isomorphism is direct product-closed
 * Direct product is cancellative for order statistics-equivalence
 * Unique factorization does not hold up to order statistics-equivalence