= Solution
The inductive definition of <ordinal addition> is
$$
\alpha+0=\alpha,\qquad
\alpha+(\beta+1)=(\alpha+\beta)+1,\qquad
\alpha+\lambda=\sup_{\beta<\lambda}(\alpha+\beta)
$$
for a <limit ordinal> $\lambda$.
The synthetic definition takes the <order type> of the <disjoint union> of a copy of $\alpha$ followed by a copy of $\beta$: every point of the first copy precedes every point of the second, and each copy retains its original <total order>.
Fix $\alpha$ and use <transfinite induction> on $\beta$. The synthetic sum with the <empty order> is $\alpha$. Appending a <greatest element> produces the <successor ordinal> rule. At a <limit ordinal> $\lambda$, the second copy is the union of its <initial segments> of types $\beta<\lambda$, so the whole order has type
$\sup_{\beta<\lambda}(\alpha+\beta)$. Thus the synthetic operation satisfies the inductive recursion; uniqueness in <transfinite recursion> proves the definitions equivalent.
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