Transplant (Abstract Algebra)/Examples/Addition on Positive Reals under Squaring

Example of Transplant
Let $\struct {\R_{>0}, +}$ be the set of strictly positive real numbers under addition.

Let $f: \R_{>0} \to \R_{>0}$ be the permutation defined as:
 * $\forall x \in \R_{>0}: \map f x = x^2$

The transplant $\oplus$ of $+$ under $f$ is given by:
 * $x \oplus y = x + y + 2 \sqrt {x y}$

Proof
From Restriction of Real Square Mapping to Positive Reals is Bijection, $f$ is a bijection.

The inverse of $f$ is given as:
 * $\forall x \in \R_{>0}: \map {f^{-1} } x = \sqrt x$

Hence from the Transplanting Theorem: