Legs of Pythagorean Triangle used as Generator for another Pythagorean Triangle

Theorem
Let $a$ and $b$ be the legs of a Pythagorean triangle $P_1$.

Let $\left({a, b}\right)$ be used as the generator for a new Pythagorean triangle $P_2$.

Then the hypotenuse of $P_2$ is the square of the hypotenuse of $P_1$.

Proof
By Pythagoras's Theorem, the square of the hypotenuse of $P_1$ is $a^2 + b^2$.

By Solutions of Pythagorean Equation, the sides of $P_2$ can be expressed as $\left({2 a b, a^2 - b^2, a^2 + b^2}\right)$, where the hypotenuse is $a^2 + b^2$.