Prove the Generalized Associative Law for Groups (i.e. a finite sum of elements can be bracketed in any way). The proof is outlined in D & F. I just want to know whether or not one part of my proof is correct. Show that for any group G under the operation °, and elements a1,...,an, any bracketing of the expression a1°...°an can be rewritten as (a1°...°ak)°(ak+1°...°an). Originally, I considered defining a directed set and working with that, but this seemed like extra unecessary steps. So, I will instead proceed by starting with the expression a1°...°an and bracketing as follows. First place brackets such that every bracket contains exactly two elements of the group and one °. Then proceed by replacing each bracketed item with a b (so replace ai with bi). Then repeat step one. Continue this process, replacing each new bracketed pair with a new indicator. It must terminate when there are two elements left because there are finite elements and operations. Then we are left with xi ° yi. So if we substitute back in all the ai along with the brackets, we will get (a1°...°ak)°(ak+1°...°an), where each half is bracketed in some way. Then we can see that the construction I've described defines all possible ways to bracket the expression, so we are done. Is this valid, and would we need to show this indeed defines all bracketing possibilities, or is it clear enough?