In the preceding posts, I have discussed the** Axioms of Real Numbers** and the **Axioms of Equality**. We will use these axioms to prove several basic theorems in Algebra. In this post, we prove the cancellation law of addition. That is, for any numbers *a*, *b*, and *c*, if *a *+ *c* = *b* + *c*, then *a* = *b*.

**Theorem**

For any real numbers *a*, *b*, *c*, if *a* + *c* = *b* + *c*, then *a* = *b*.

**Proof**

*a* + *c* = *b* + *c*

Adding –*c* to both sides, we have

(*a* + *c*) + (-*c*) = (*b* + *c*) + (-*c*)

Using *Axiom 2A*, the associativity of addition, we have

*a* + (*c* + (-*c*)) = *b* + (*c* + (-*c*)).

By* Axiom 6A*, the existence of additive inverse, we get

*a* + 0 = *b* + 0.

By *Axiom 5A*, the existence of additive identity

*a* = *b*.

That ends the proof.