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 $$\phi \left ({x}\right) = e^{\imath x}$$.

Then $$\phi$$ is a homomorphism.

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

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