= Solution
The inductive definition of <ordinal addition> is $\alpha+0=\alpha$, $\alpha+(\beta+1)=(\alpha+\beta)+1$, and $\alpha+\lambda=\sup_{\beta<\lambda}(\alpha+\beta)$ for a nonzero limit ordinal $\lambda$. The synthetic definition is the <order type> of the <order sum> obtained by putting a copy of $\beta$ after a disjoint copy of $\alpha$.
These agree by <transfinite induction> on the second argument. Appending no points gives $\alpha$; appending one new greatest point gives a successor. At a limit stage, the order sum is exhausted by the increasing initial segments obtained by appending each $\beta<\lambda$, so its order type is their supremum. Thus the synthetic construction satisfies all three recursion clauses, and uniqueness of transfinite recursion proves equivalence.
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