Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 2 / 16F / i

This assertion is always true. Synthetically, either bracketing places three disjoint blocks of order types $\alpha ,\beta ,\gamma$ in that order. The identity on these blocks is an order isomorphism, so both sums have the same order type. This proves associativity of ordinal addition for successor and limit ordinals alike.