Infinite Cyclic Group is Unique up to Isomorphism

Theorem
All infinite cyclic groups are isomorphic.

That is, up to isomorphism, there is only one infinite cyclic group.

Proof
Let $G_1$ and $G_2$ be infinite cyclic groups.

From Infinite Cyclic Group is Isomorphic to Integers we have:
 * $G_1 \cong \left({\Z, +}\right) \cong G_2$

where $\left({\Z, +}\right)$ is the additive group of integers.

From Isomorphism is Equivalence Relation it follows that:
 * $G_1 \cong G_2$

Comment
Now that as we have, in a sense, defined an infinite cyclic group with reference to the additive group of integers that we painstakingly constructed in the definition of integers, it naturally follows that we should use $\left({\Z, +}\right)$ as an "archetypal" infinite cyclic group.