# Proof:⟦a⟧=⟦b⟧ if and only if aRb

Proof:
Suppose first that ; we will show that . Since is reflexive, we have . Thus . Since , this means , which by definition means , as required.

Now, suppose aRb; we will show that . Since , we have . Since is reflexive, we have so . Since equivalence classes form a partition, and and overlap, we conclude that , as required.

Note that you can prove this result without resorting to the fact that equivalence classes form a partition, but you would just be repeating a lot of the work done in that proof.