Exponential of Sum/Real Numbers

Theorem
Let $$x, y \in \mathbb{R}$$ 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
Let $$X = \exp x$$ and $$Y = \exp y$$.

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

From the Basic Properties of Exponential Function, $$\exp \left({\ln x}\right) = x$$.

Thus $$\exp \left({x + y}\right) = \exp \left({\ln XY}\right) = XY = \left({\exp x}\right) \left({\exp y}\right)$$.

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