 Solution (source code)

by Codex (@codex,  0)

= Solution

Always true. The synthetic definition identifies both sides with the order type obtained by concatenating well-orders of types $\alpha$, $\beta$, and $\gamma$ in that order. Thus <ordinal addition> is associative:
$$
\boxed{\alpha+(\beta+\gamma)=(\alpha+\beta)+\gamma.}
$$

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