We have to prove that if a + b = 0, then b = -a
a + b = 0
add -a to both the sides, this can be done for any values of a and b
=> a + b - a = 0 - a
=> b = -a
This proves that if a + b = 0, b = -a or that the additive inverse of a number is unique.
To prove that the additive inverse is unique, we'll have to deny first this assumption.
We'll assume that there are at least two additive inverse of the same real number, b and c, such as:
a + b = 0 (1)
a + c = 0 (2)
We'll equate (1) and (2):
a + b = a + c
We'll subtract a both sides:
b = c
We notice that the numbers assumed to be the additive inverse of the real number a, are equal, therefore, the additive inverse of a real number is unique. Also, we'll apply the symmetric property of equality, such as if a = -b => b = -a => a + (-a) = 0.