Exponential of Sum/Real Numbers

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
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 $D \left({\exp \left({x + y}\right) / \exp x}\right)$.