(-1)a + a = (-1)a + (1)a ;Multiplicative Identity Property
=(-1 + 1)a ;Distributive Property
= 0 ;Multiplication Property Of Zero
This means (-1)a is an additive inverse if a ;Cancellation Property Of Addition
Since the additive inverse is unique (-1)a = -a
By the way you can't start a proof by writing what you want to prove first!! (In this case you can't write (-1)a = -a as the first line of your proof. It comes last!!)