  Let $L$ be an $n\times n$ array whose top left $r\times s$ subarray is filled
with $k$ different symbols, each occurring at most once in each row and at most
once in each column. We find necessary and sufficient conditions that ensure
the remaining cells of $L$ can be filled such that each symbol occurs at most
once in each row and at most once in each column, and each symbol occurs a
prescribed number of times in $L$. The case where the prescribed number of
times each symbol occurs is $n$ was solved by Ryser (Proc. Amer. Math. Soc. 2
(1951), 550--552), and the case $s=n$ was settled by Goldwasser et al. (J.
Combin. Theory Ser. A 130 (2015), 26--41). Our technique leads to a very short
proof of the latter.
