The proof of this magic trick is relatively easy, check it out.

Given a two digit number

So ab = 10*a + b

Now subtract a + b from ab which gives you 10a + b – ( a + b) = 9a

By the divisibility rule for 9, the sum of the digits will be a multiple of 9

Since 10 < a < 0 the sum of 9a will be 9