Group/Examples/x+y over 1+xy/Isomorphic to Real Numbers/Proof 2

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:
 * $\forall x, y \in G: x \circ y = \dfrac {x + y} {1 + x y}$

Proof
To prove $G$ is isomorphic to $\struct {\R, +}$, it is sufficient to find a bijective homorphism $\phi: \to \R \to G$:


 * $\forall x, y \in G: \map \phi {x + y} = \map \phi x \circ \map \phi y$

From Group Examples: $\dfrac {x + y} {1 + x y}$:
 * the identity element of $G$ is $0$
 * the inverse of $x$ in $G$ is $-x$.

This also holds for $\struct {\R, +}$.

This hints at the structure of a possible such $\phi$:
 * $(1): \quad$ that it is an odd function
 * $(2): \quad$ that it passes through $0$
 * $(3): \quad$ that its image is the open interval $\openint {-1} 1$.

One such function is the hyperbolic tangent function $\tanh$, which indeed has the above properties on $\R$.

Then we have that:

This demonstrates the homorphism between $\struct {\R, +}$ and $\struct {G, \circ}$.

We have that Real Hyperbolic Tangent Function is Strictly Increasing over $\R$.

Hence from Strictly Monotone Real Function is Bijective, $\tanh: \R \to \openint {-1} 1$ is a bijection.

Hence the result that $\struct {\R, +} \cong \struct {G, \circ}$.