Everyone knows that a * (b * c) = (a * b) * c
But how to prove the same rule for more factors? I know how to do this for 4, 5 factors separately. But how to prove right away that any arrangement of brackets in an expression from any number of factors is true?
Induction.
For three elements it doesn't matter how you place brackets. That's the base case.
Assume that for any combination of $k$ or fewer terms, it doesn't matter how you place brackets. Now take $k+1$ arbitrary elements, and place brackets in two different ways.
By the induction hypothesis, only the outermost pair of brackets could influence the value (anything inside the outermost brackets is $k$ or fewer terms). So, what we have to compare are the values of the two expressions $$ (a_1\cdots a_i)(a_{i+1}\cdots a_{k+1})\\ (a_1\cdots a_j)(a_{j+1}\cdots a_{k+1}) $$ for some $1\leq i<j\leq k$. Now note that we have $$ (a_1\cdots a_i)((a_{i+1}\cdots a_j)(a_{j+1}\cdots a_{k+1}))=((a_1\cdots a_i)(a_{i+1}\cdots a_j))(a_{j+1}\cdots a_{k+1}) $$ by associativity on three elements.
