Direct product preserves powering

Statement for direct product of two groups
Suppose $$G_1$$ and $$G_2$$ are groups both of which are powered over a set $$\pi$$ of prime numbers. Then, the external direct product $$G_1 \times G_2$$ is also powered over the set $$\pi$$.

Statement for direct product of finitely many groups
Suppose $$G_1,G_2,\dots,G_n$$ is a collection of finitely many groups all of which are powered over a set $$\pi$$ of prime numbers. Then, the external direct product $$G_1 \times G_2 \times \dots \times G_n$$ is also powered over the set $$\pi$$.

Statement for infinite direct products
Suppose $$G_i, i \in I$$ is a collection of groups all of which are powered over a set $$\pi$$ of prime numbers. Then:


 * The (unrestricted) external direct product of all the $$G_i$$s is also powered over the set $$\pi$$.
 * The restricted external direct product of the $$G_i$$s is also powered over the set $$\pi$$.