Product with Repdigit can be Split into Parts which Add to Repdigit

Theorem
Let $n$ be a positive integer with $d_1$ digits.

Let $m$ be a repdigit number with $d_2$ digits such that $d_2 > d_1$.

Let $r$ consist of the result when the rightmost $d_2$ digits of $m n$ is cut off and added to the remaining left hand portion.

Then $r$ is a repdigit number.