Submagma of group is cancellative semigroup

Statement
Suppose $$G$$ is a group and $$S$$ is a subset of $$G$$ that is closed under the multiplication of $$G$$. Then, $$S$$ is a cancellative semigroup.

Facts used

 * 1) uses::Invertible implies cancellative in monoid, which tells us that the cancellation property holds in groups.

Proof
The proof follows from Fact (1), and the fact that cancellation is inherited on subsets.