Equivalence of Definitions of Real Exponential Function/Extension of Rational Exponential implies Differential Equation

Theorem
The following definition of the concept of the real exponential function:

As an Extension of the Rational Exponential
implies the following definition:

Proof
Let $\exp x$ be the real function defined as the extension of rational exponential.

Then we have:

The application of Derivative of Exponential at Zero is not circular as the referenced proof does not depend on $D_x \exp x = \exp x$.