Exponential of Sum/Real Numbers/Proof 4

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 real function $f: \R \to \R$ defined by:
 * $\map f x := \dfrac {\map \exp {x + y} } {\map \exp y}$

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: