Cyclic Permutation of Kaprekar Number

Theorem
Let $n$ be a Kaprekar number of $k$ digits.

Let $m$ be an integer formed from a cyclic permutation of the digits of $n$.

Let $m$ be squared and the result split into $2$ parts, where the $2$nd part is of $k$ digits.

Let these two parts be added, in the way of operating on a Kaprekar number.

If the result is more than $k$ digits long, split that into $2$ parts, where the $2$nd part is of $k$ digits, and add the parts.

The result will be another cyclic permutation of the digits of $n$.