Epimorphism Preserves Semigroups

Theorem
Let $\phi: \left({S, \circ}\right) \to \left({T, *}\right)$ be an epimorphism.

If $\left({S, \circ}\right)$ is a semigroup, then so is $\left({T, *}\right)$.

Proof
If $\left({S, \circ}\right)$ is a semigroup, then by definition it's closed.

From Morphism Property Preserves Closure, $\left({T, *}\right)$ is therefore also closed.

If $\left({S, \circ}\right)$ is a semigroup, then by definition $\circ$ is associative.

From Epimorphism Preserves Associativity, $*$ is therefore also associative.

So $\left({T, *}\right)$ is closed, and $*$ is associative, and therefore by definition, $\left({T, *}\right)$ is a semigroup.