Two sets having the same subset sums.

Your basic approach of induction on the number of elements in the multisets is a good one. I think you can simplify it by saying that any $A_i$ that includes $a_1$ must have a matching $B_i$ that includes an element equal to $b_1$ because otherwise $A_i\setminus a_1$ would have no matching multiset. Now you can delete $a_1,b_1$ from any pair that have both of them and get to your claim that $A\setminus a_1$ and $B \setminus b_1$ have mathching subset sums.

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Combinatorics

Summation

Additive Combinatorics

Elementary Number Theory

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