Isomorphism Preserves Semigroups/Proof 1

Proof
If $\struct {S, \circ}$ is a semigroup, then by definition it is closed.

From Morphism Property Preserves Closure, $\struct {T, *}$ is therefore also closed.

If $\struct {S, \circ}$ is a semigroup, then by definition $\circ$ is associative.

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

So $\struct {T, *}$ is closed, and $*$ is associative, and therefore by definition, $\struct {T, *}$ is a semigroup.