Exponential of Sum/Real Numbers/Proof 2

Lemma
This proof assumes the definition of $\exp$ as defined by a limit of a sequence:


 * $\exp x = \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$

From Powers of Group Elements we can presuppose the Exponent Combination Laws for natural number indices.

First we introduce a lemma:

By definition: