Exponential of Sum/Real Numbers/Proof 4

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 an initial value problem.

That is, suppose $\exp$ satisfies:
 * $(1): \quad D_x \exp x = \exp x$
 * $(2): \quad \exp 0 = 1$

on $\R$.

Consider the mapping $f : \R \to \R$ defined by:
 * $f \left({ x }\right) := \dfrac{ \exp \left({ x + y }\right) }{ \exp \left({ y }\right) }$

From Exponential of Real Number is Strictly Positive, $f$ is well-defined.

So:

Thus $f$ satisfies $(1)$.

Further:

So $f$ satisfies $(2)$.

From Exponential Function is Well-Defined, $f = \exp$.

That is: