Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers

Theorem
Let $G = \set {x \in \R: -1 < x < 1}$ be the set of all real numbers whose absolute value is less than $1$.

Let $\circ: G \times G \to G$ be the binary operation defined as:
 * $\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$

It has been established in Group Examples: $\dfrac {x + y} {1 + x y}$ that $\struct {G, \circ}$ forms a group.

$\struct {G, \circ}$ is isomorphic to the additive group of real numbers $\struct {\R, +}$.