Exponent Combination Laws/Product of Powers/Proof 2

Theorem
Let $a \in \R_{> 0}$ be a positive real number.

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

Let $a^x$ be defined as $a$ to the power of $x$.

Then:
 * $a^x a^y = a^{x + y}$

Proof
Let $x, y \in \R$.

From Rational Sequence Decreasing to Real Number, there exist rational sequences $\left\langle{x_n}\right\rangle$ and $\left\langle{y_n}\right\rangle$ converging to $x$ and $y$, respectively.

Then, since Power Function on Strictly Positive Base is Continuous: Real Power: