Exponential of Sum/Real Numbers/Proof 2

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 defined by a limit:

$\exp x = \displaystyle \lim_{n \to +\infty} \left({1 + \frac x n}\right)^n$

Note that from Powers of Group Elements we can presuppose the exponent combination laws for natural number indices.

By definition: