 Solution

A reduced word is either the empty word or a productin which every syllable $u_i$ is a nonidentity element of $G$ or $H$, and consecutive syllables belong to different factors.

Let $\mathcal{W}$ be the set of reduced words. Each $g\in G$ acts on the right of $\mathcal{W}$: if the last syllable lies in $H$, append $g$; if it lies in $G$, multiply it by $g$ and delete it when the product is the identity. Define the action of each $h\in H$ analogously. These rules give genuine actions of the two factors by permutations of $\mathcal{W}$, hence an action of the free group $F(X\bigsqcup Y)$. Every relation in $R\bigsqcup S$ acts trivially, so the action factors through the displayed presentation of $G\ast H$.

The element represented by a reduced word $w$ sends the empty word to $w$. Therefore two reduced words representing the same element induce the same permutation and have the same value on the empty word. They must be identical. This proves the normal form theorem for a free product.