Proof: By Lemma 1 from Section 4.1 we know that $z_{i}^{\prime}$ as any invertible function can be written in form $c_{i}z^{I^{(i)}}(1+o(1))$ for some nonzero $c_{i}\in K$ and a multi-index $I^{(i)}\in{\bf Z}^{n}$. Vectors $I^{(1)},\dots,I^{(n)}$ form a basis of ${\bf Z}^{n}$, as follows from the condition that $z_{1}^{\prime},\dots,z_{n}^{\prime}$ form a coordinate system. Therefore, after applying the change of coordinates $z_{i}\mapsto c_{i}z^{I^{(i)}}$ preserving form $\bigwedge_{i}dz_{i}/z_{i}$ up to sign, we may assume that $z_{i}^{\prime}=(1+o(1))z_{i}$. The Jacobian matrix of the transformation $(z_{i})\to(z_{i}^{\prime})$ is the identity matrix plus terms of size $o(1)$. Therefore its determinant has norm equal to 1. $\blacksquare$