  If $R$ and $M$ are Hilbert modules (in the sense of R. G. Douglas and V. I.
Paulsen), we study the relationship between invertible module maps $X:R\to{M}$
and $X_{z}:R/R_{z}\to{M/M_{z}}$. In particular, for quasi-free Hilbert modules
$R$ and $M$, we provide a condition of a module map $X:R\to{M}$, such that if
$X_{z}:R/R_{z}\to{M/M_{z}}$ is invertible for every $z$ in a domain $\Omega$ in
the complex plane, then $X$ is also invertible.
