KöMaL 2018 B4988

In an $(m+2)\times(n+2)$ table, we cut out the four $1\times1$ corners''. Arbitrary real numbers are written in each field of the first and last rows, and in the first and last columns of the truncated table obtained in this way. Prove that it is possible to fill in the remaining $m\times n$ interior'' of the table in a unique way with real numbers such that every number is the arithmetic mean of the four adjacent numbers.