= Solution
\b[True.] Let $G=\langle X\mid R\rangle$ and $H=\langle Y\mid S\rangle$, using disjoint finite generating alphabets. The <direct product of groups> has finite presentation
$$
G\times H=\langle X,Y\mid R,S,[x,y]=1\ (x\in X,\ y\in Y)\rangle.
$$
The commuting relations let every word be reordered as a word in $X$ followed by a word in $Y$; the two projections then show that no additional relations within either factor are imposed. For an input word $w$, form its projections $w_X$ and $w_Y$ by deleting all letters from the other factor. Evaluate each projection with the given <word problem for a group> algorithm for that factor. The product coordinates give
$$
\boxed{w=1\text{ in }G\times H\quad\Longleftrightarrow\quad w_X=1\text{ in }G\ \text{and}\ w_Y=1\text{ in }H.}
$$
Both component algorithms halt, so this is a terminating algorithm. No subgroup-membership procedure or comparison between unknown presentations is required.
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