Transplant (Abstract Algebra)/Examples/Multiplication on Reals under 1-x

Example of Transplant
Let $\struct {\R, \times}$ be the set of real numbers under multiplication.

Let $f: \R \to \R$ be the permutation defined as:
 * $\forall x \in \R: \map f x = 1 - x$

Then the transplant $\otimes$ of $\times$ under $f$ is given by the
 * $x \otimes y = x + y - x y$

Proof
The fact that $f$ is a bijection is taken as read.

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

Hence from the Transplanting Theorem: