Isomorphism Preserves Semigroups/Proof 1

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