Infinite Cyclic Group is Unique up to Isomorphism

Corollary to Infinite Cyclic Group Isomorphic to Integers
All infinite cyclic groups are isomorphic.

Or, 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 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.