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(A,*) and (B,[tex]\nabla[/tex])

So to prove associativity, since I know that both A and B are groups, their direct product will be a group. Could I do the following

a

_{i}, b

_{i}[tex] \in A,B

[/tex]

[(a

_{1},b

_{1})(a

_{2},b

_{2})](a

_{3},b

_{3})=(a

_{1},b

_{1})[(a

_{2},b

_{2})(a

_{3},b

_{3})]

Since A and B are groups, I know they have distributing everything via the proper binary operations (I got kind of lazy at this point). Can I just multiply both sides by an a

^{-1}

_{1}b

^{-1}

_{1}and so on until i get something like b

_{3}=b

_{3}

I just want to make sure this is sort of hte process one uses to prove the direct product is a group. By the way this question comes from Dummit and Foote, it says that the proof of this is left as a straightforward excercise.