Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 129 / 1 / i / Solution

For every $x\in B{C}^{-1}$, choose one representation $x=b_x{c}_x^{-1}$. DefineThe image determinesand the chosen representative then determines $a=(a{b}_x^{-1})b_x$. Thus $F$ is injective, and counting its domain and codomain proves the Noncommutative Ruzsa triangle inequality