Exponential Function is Well-Defined/Real/Proof 4

Theorem
Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then $\exp x$ is well-defined.

Proof
This proof assumes the definition of the exponential as the inverse of the logarithm.

Let:
 * $k \in \N$
 * $J_k = \left[{\dfrac 1 k \,.\,.\, k}\right]$
 * $J_k^o = \left({\dfrac 1 k \,.\,.\, k}\right)$

and let $\ln_k$ denote the restriction of $\ln$ to $J_k^o$.

Lemma
From Union of Functions Theorem, $\displaystyle \bigcup \left\{ {\ln_k: k \in \N}\right\} = \ln$.

From Natural Logarithm Function is Continuous, $\ln$ is continuous on $J_k$.

Also:

So from Derivative of Inverse Function, $\ln_k$ is invertible.

From Union of Functions Theorem: Corollary, $\ln = \displaystyle \bigcup \left\{ {\ln_k: k \in \N}\right\}$ is invertible.

Hence the result, from Inverse Mapping is Unique.