Prove: If a + b = 0 then b = -a. (This states that the additive inverse of a real 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.