Quotients of elementarily equivalent abelian groups by multiples of n are elementarily equivalent

Statement
Suppose $$G$$ and $$H$$ are fact about::abelian groups that are fact about::elementarily equivalent groups. Then, the quotient groups $$G/nG$$ and $$H/nH$$ are also elementarily equivalent. Here:

$$nG := \{ ng \mid g \in G \}, \qquad nH := \{ nh \mid h \in H \}$$.

Proof
We want to show that we can establish an elementary local isomorphism between $$G/nG$$ and $$H/nH$$. We do this in terms of the language of two-player games, where the prover is trying to establish that the two are elementarily equivalent and the disprover is trying to establish that the two are not elementarily equivalent.

Here's what we do. For every element that the disprover picks in $$H/nH$$, the prover goes to the game for $$G$$ and $$H$$ and imagines that the disprover picked there an element in $$H$$ in that coset. The prover then sees what element to pick in the $$G-H$$ game, and picks the coset of that element downstairs. The prover follows this strategy throughout the game.

Now, we need to show that this strategy works. For this, note that if, at any stage in the $$G/nG - H/nH$$ game, the disprover wins, then the disprover has a word in one structure that is zero such that the corresponding word in the other structure isn't, then at the corresponding stage in the $$G - H$$ game, the disprover has a word in one structure that yields a multiple of $$n$$ but the corresponding word in the other structure isn't. But this translates to the disprover winning the $$G-H$$ game, a contradiction.