This is the second to the last post on proofs of the Basic Algebra Theorems. In this post, we prove that -(*a* + *b*) = (-*a*) + (-*b*).

**Theorem**

For any *a* and *b*, -(*a* + *b*) = (-*a*) + (-*b*).

**Proof**

– (*a* + *b*) = (-1)(*a* + *b*) by Corollary 1 of this theorem on the product of positive and negative numbers.

By the Distributive Property (Axiom 4), we have (-1)(*a* + *b*) = (-1)(*a*) + (-1)(*b*).

(-1)(*a*) + (-1)(*b*) = (-*a*) + (-*b*) again, by Corollary 1 of this theorem.

This ends the proof.