Integer which is Multiplied by 9 when moving Last Digit to First

Theorem
There is one and only one positive integer $N$ such that if you move the last digit to the front, the result is the positive integer $9 N$:
 * $10 \, 112 \, 359 \, 550 \, 561 \, 797 \, 752 \, 808 \, 988 \, 764 \, 044 \, 943 \, 820 \, 224 \, 719$

Proof
First we demonstrate that this number has this property: