Exponential of Sum/Real Numbers/Proof 1

Theorem
Let $x, y \in \R$ be real numbers.

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

Then:
 * $\exp \left({x + y}\right) = \left({\exp x}\right) \left({\exp y}\right)$

Proof
This proof assumes the definition of $\exp$ as:


 * $\exp x = y \iff \ln y = x$,


 * $\ln y = \displaystyle \int_1^y \frac 1 t \ \mathrm dt$

Let $X = \exp x$ and $Y = \exp y$.

From Sum of Logarithms, we have:
 * $\ln X Y = \ln X + \ln Y = x + y$

From the Exponential of Natural Logarithm:
 * $\exp \left({\ln x}\right) = x$

Thus:
 * $\exp \left({x + y}\right) = \exp \left({\ln X Y}\right) = X Y = \left({\exp x}\right) \left({\exp y}\right)$

Alternatively, this may be proved directly by investigating:
 * $D \left({\exp \left({x + y}\right) / \exp x}\right)$