Answer
According to theorem 2, additive inverses are unique. If we say for all real numbers x and y, (−x) · y =x · (−y) = −(x · y), then their additive inverses are the same.
Work Step by Step
Lemma: the additive inverse of -(x.y) is (x.y).
If -(x.y) = (-x).y, then:
(-x).y + (x.y) = 0 (according to inverse additive)
y(-x+x) = 0 (according to distributive law)
y.0 = 0 (according to inverse additive)
0 = 0 (according to theorem 5)
Conclusion: (-x).y is the inverse additive of (x.y) therefore according to theorem 2 (-x).y = -(x.y)
If -(x.y) = x(-y), then:
x.(-y)+ (x.y) = 0 (according to inverse additive)
x(-y+y) = 0 (according to distributive law)
x.0 = 0 (according to inverse additive)
0 = 0 (according to theorem 5)
Conclusion: x.(-y) is the inverse additive of (x.y) therefore according to theorem 2 x.(-y) = -(x.y)
Last conclusion: −(x · y) = (−x) · y = x · (−y)
*disclaimer: this solution is not absolute, please modify it to your imagination
