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I am trying to prove the generalized associative law with induction, but am being tripped up by one aspect. I am reading a solution and it says for the induction step argue that any bracketing of the product ##a_1 \cdot a_2 \cdot \cdots a_n## must break into two subproducts ##(a_1 \cdot \cdots \cdot a_k) \cdot (a_{k+1} \cdot \cdots \cdot a_n)## where each subproduct is bracketed in some fashion. Why is this true? As a concrete example, what are the subproducts of ##(a_1 \cdot (a_2 \cdot (a_3 \cdot a_4))) \cdot a_5##, where the first product as two elements and the second has three, for example?