The Proof of the Cancellation Law of Addition

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.

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