Exponential of Sum/Real Numbers/Proof 1

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


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

where:


 * $\ln y = \displaystyle \int_1^y \dfrac 1 t \rd t$

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:
 * $\map \exp {\ln x} = x$

Thus:
 * $\map \exp {x + y} = \map \exp {\ln X Y} = X Y = \paren {\exp x} \paren {\exp y}$

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