Prove that if G is a set with an associative operation which has the solvability and cancellation properties, then it has the identity and inverse properties.
Solvability: for any elements , of , there is an element in with , and an element in with
Cancellation: for any three elements of , if , then
Existence of Identity: there is an element of so that for all in ,
Existence of Inverse: for any in , there is an element in so that