Equivalence of Definitions of Real Exponential Function/Proof 2

Theorem
The following definitions of the exponential function are equivalent.

Proof
From Derivative of Exponential Function and Exponential of Zero, each definition of $\exp$ satisfies the following:
 * $(1): \quad D_x \exp = \exp$
 * $(2): \quad \map \exp 0 = 1$

on $\R$.

From Exponential Function is Well-Defined, such a solution is unique.

Thus they all are all equivalent.