KöMaL 2010 C1045

Three regular dice are rolled. The three numbers obtained on top are listed in an arbitrary order. Then the list is continued with the numbers obtained on the bottom, in the same order. The resulting six-digit number is divided by 111, 7 is subtracted from the ratio, and the difference is divided by 9. Prove that the result is a three-digit number whose digits are the numbers rolled.