We define a system with the domain of all integers and a new operation (named # for the purpose of this proof), the function being: a # b = 2 - a + b.

plug in 2 and 1, and you get 2 # 1 = 2 - 2 + 1 = 1. By abuse of notation, we can rename # as + in our new system, so 2 + 1 = 1.

The whole point of our new system and operation is for 2 to be the neutral element in regard to the operation.

Edit: I took a closer look at this and realized that 2 is not a neutral element for this operation. This operation just happens to work.