Let G be a set and let * be a binary operation.
Suppose that ∃e∈G, such that e * a = a, ∀a∈G
and ∃a'∈G, such that a' * a = e, ∀a∈G
Note that e * e  = e. (a' * a) * e = a' * a
a * (a' * a) * e = a * (a' * a)
(a * a') * a * e = (a * a') * a
e * a * e = e * a
a * e = a (By cancellation law for e)

a' * a = e, (a' * a) * a' = e * a' = a'
a' * (a * a') = a'
a * a' = e (By cancellation law for a')

Thus G is a group