Exponential Function is Well-Defined/Real/Proof 5

Proof
This proof assumes the definition of $\exp$ as the solution to an initial value problem.

That is, suppose $\exp$ solves:
 * $(1): \quad \dfrac \d {\d x} y = \map f {x, y}$
 * $(2): \quad \map \exp 0 = 1$

on $\R$, where $\map f {x, y} = y$.

From Derivative of Exponential Function: Proof 4, the function $\phi : \R \to \R$ defined as:
 * $\ds \map \phi x = \sum_{k \mathop = 0}^\infty \frac {x^k} {k!}$

satisfies $\map {\phi'} x = \map \phi x$.

So $\phi$ satisfies $(1)$.

From Exponential of Zero: Proof 3:
 * $\map \phi 0 = 1$

So $\phi$ satisfies $(2)$.

Thus, $\phi$ is a solution to the initial value problem given.

From Exponential Function is Continuous: Proof 5 and $(1)$:
 * $\phi$ is continuously differentiable on $\R$.

Because $\map f {x, \phi} = \phi$:
 * $f$ is continuously differentiable on $\R^2$.

Thus, from Uniqueness of Continuously Differentiable Solution to Initial Value Problem, this solution is unique.