I'm currently reading "Quantum Groups", Christian KASSEL, 1995
In chapter XVIII. section 8, he is proving the second rigidity theorem for quantum enveloping algebras, namely if $ A = (U(\mathfrak{g})[[h]], \Delta, \epsilon, \phi, R)$ and $A' = (U(\mathfrak{g})[[h]], \Delta, \epsilon, \phi', R)$ are quantum enveloping algebra for the same finite dimensional semisimple Lie algebra $\mathfrak{g}$, with $R_{21} = R$ and $ R \equiv 1 \otimes 1 $ modulo $h$, and $R = \Delta(u)V$ with $V$ central element, then $\phi$ and $\phi'$ are related by some gauge transformation.
Here, $\phi$ and $\phi'$ are elements of $U(\mathfrak{g})^{\otimes 3}[[h]]$ (formal series)
At some point in the proof, he defines $\varphi$ in $U(\mathfrak{g})^{\otimes 3}$ such that $\phi' = \phi + h^{n}\varphi $ modulo $h^{n+1}$.
And then uses the relation : $(\phi + h^{n}\varphi)^{-1} \equiv \phi - h^{n}\varphi$ modulo $h^{n+1}$.
Where does this last relation come from please?
Dearly