7. Given the inverse function of $y=f(x+1)$ is
$$
\begin{array}{c}
y=f^{-1}(x+1) \text {, and } f(1)=4007 \text {. Then } \\
f(1998)=
\end{array}
$$
II, 7.2010.
From $y=f^{-1}(x+1)$, we get $x+1=f(y)$, which means $x=f(y)-1$.
Thus, the inverse function of $y=f^{-1}(x+1)$ is $y=f(x)-1$. Therefore, $f(x+1)-f(x)=-1$.

Let $x=1,2, \cdots, 1997$, add up all the equations and simplify to get $f(1998)-f(1)=-1997$.
Hence, $f(1998)=2010$.