Isomorphism Preserves Cancellability

Theorem
Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

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

Then:
 * $a \in S$ is cancellable in $\left({S, \circ}\right)$ iff $\phi \left({a}\right) \in T$ is cancellable in $\left({T, *}\right)$.

Proof 1
Let $\left({S, \circ}\right)$ be an algebraic structure in which $a$ is cancellable.

From Isomorphism Preserves Left Cancellability and Isomorphism Preserves Right Cancellability:
 * $a \in S$ is left cancellable in $\left({S, \circ}\right)$ iff $\phi \left({a}\right) \in T$ is left cancellable in $\left({T, *}\right)$

and
 * $a \in S$ is right cancellable in $\left({S, \circ}\right)$ iff $\phi \left({a}\right) \in T$ is right cancellable in $\left({T, *}\right)$.

It follows that:
 * $a \in S$ is cancellable in $\left({S, \circ}\right)$

iff
 * $\phi \left({a}\right) \in T$ is cancellable in $\left({T, *}\right)$.