Answer
According to theorem 2, inverse additives are unique. If we say for all real numbers x and y, −(x + y) =(−x) + (−y), then their inverse additives are the same.
Work Step by Step
Lemma: The inverse additive of -(x+y) is (x+y)
if -(x+y) = (-x)+(-y), then:
(-x)+(-y) + (x+y) = 0 (according to inverse additive)
(-x) + (x+y) + (-y) = 0 (according to commutative law)
(-x+x) + (y+(-y)) = 0 (according to associative law for addition)
0 + (y+(-y)) = 0 (according to inverse additive)
0 + 0 = 0 (according to inverse additive)
0 = 0 (according to additive identity law)
Conclusion: (-x) + (-y) is the inverse additive of (x+y) therefore (-x) + (-y) = -(x+y)
