How do you prove the left and right side of an identity of a set are equal? I'm having some trouble understanding sets w/ associative binary operations. Say I have a set "S" w/ the associative binary operation SxS -> S. If 'L' is a left identity of S and 'R' is a right identity of S, how can I prove those two are equal?
The definition of associativity is for all a, b, c in G, we have (a * b) * c = a * (b * c). But how does that apply to proving they are equal? It seems obvious but I'm sure there's a certain "proof" way of doing it?

Well this is the proof. 
$$ L =  L * R = R$$
No need to use associativity. 

For an elaboration. 
A left identity is an element $L$ satisfying $L * g = g$ for every $g \in G$. Hence $L * R = R$. 
A right identity is an element satisfying $g * R = g \;\; \forall g \in G$. Hence, $L * R = L$. 
