Updated 2018 November 17
A group is a set S with an operation + and an identity element 0 such that:
By additive inverse property, there exists a -x with (-x)+x=0. Likewise by the additive inverse property there exists a -(-x) with -(-x)+(-x)=0. Then adding -(-x) to both sides of the first equation, -(-x)+(-x)+x=-(-x), implies x=-(-x).