Homomorphism from Reals to Circle Group

Theorem
Let $\left({\R, +}\right)$ be the additive group of real numbers.

Let $\left({K, \times}\right)$ be the circle group.

Let $\phi: \left({\R, +}\right) \to \left({K, \times}\right)$ be the mapping defined as $\phi \left ({x}\right) = e^{i x}$.

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

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