Homomorphism from Reals to Circle Group

Theorem
Let $\struct {\R, +}$ be the additive group of real numbers.

Let $\struct {K, \times}$ be the circle group.

Let $\phi: \struct {\R, +} \to \struct {K, \times}$ be the mapping defined as:
 * $\forall x \in \R: \map \phi x = e^{i x}$

Then $\phi$ is a (group) homomorphism.

Proof
Let $x, y \in \R$.

Then: