Intrigued, I decided to try to establish the general case and then prove it using mathematical induction. As it turns out, the general case is so incredibly trivial that constructing a proof would be absurd. (Indeed, it resolves to an algebraic tautology).
BUT – this exercise didn’t satisfy me because it didn’t capture the idea of “adding the numbers represented by the individual numerals composing a number”. It occurred to me that in order to do this, one would have to construct a model (c.f. Gödel) of Peano arithmetic that gives you a metalanguage to mobilize the distinction between “numeral” (or digit) and “number”. Too long and too hard for me (I’m not very good at logic nor math).
If you did make such a model, then I suspect you could also generalize THAT case to all possible base-10 positional notation systems (other than the Hindu-Arabic numeral system).
Finn: As a historian of science I’m sadly pretty lackluster in the math department…But I do find it interesting that the “model” to “capture the idea of ‘adding the numbers represented by individual numeral composing a number’” would be a little complex. It’s an axiom of numerology to employ that process (until you pare down to a “pure” single digit). Demystifying it would be fascinating…And terribly bothersome to devoted numerologists…