 Noncommutative Ruzsa triangle inequality

For nonempty finite subsets $A,B,C$ of an arbitrary group,Choose one representation $x=b_x{c}_x^{-1}$ for each $x\in B{C}^{-1}$. The map $(a,x)\mapsto (a{b}_x^{-1},a{c}_x^{-1})$ is injective.