Exponential of Sum/Real Numbers/Proof 5

Theorem
Let $x, y \in \R$ be real numbers.

Let $\exp x$ be the exponential of $x$.

Then:
 * $\exp \left({x + y}\right) = \left({\exp x}\right) \left({\exp y}\right)$

Proof
This proof assumes the definition of $\exp$ as a series.

Then: