Chapter 13 - Appendix 1 - Axioms for the Real Numbers and the Positive Integers - Exercises - Page A-6: 5

We can proof the statement by introducing new variables without changing the statement.

Statement: (-x).(-y) = (-x).(-y) + 0 (according to additive identitiy law) = (-x).(-y) + 0.y (according to theorem 5) = (-x).(-y) + (x-x).y (according to inverse additive law) = (-x).(-y) + x.y + (-x).y (according to distributive law) = (-x)(-y) + (-x).y + x.y (according to commutative law) = (-x).(-y+y) + x.y (according to distributive law) = (-x).0 + x.y (according to inverse additive law) = 0+ x.y (according to theorem 5) = x.y (according to additive identity law) = (x.y) Therefore we could conclude that (-x).(-y) = (x.y)