Proof that -(a + b) = (-a) + (-b)

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.

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