I agree with this comment on the question: "[...] For self answering do try to start with a question which is not already available on mathse and add good amount of context." − Paramanand Singh
The point is that Math SE is not supposed to be a blog for anyone to just post whatever they like. If you have some question that you know the answer to, and the question has not been asked before, and your answer does not seem to be available anywhere on Math SE, and you have expertise in that topic, then it makes sense to post a Q&A pair. If the answer has been posted before, it clearly should not be posted again. If the question has been asked before, you ought to post your answer there instead of contributing to the question duplicates on Math SE. If you are not sure that your answer is of high quality, then you should put it in your question and ask for feedback rather than putting it forth as a model answer.
In particular, a one-minute search yields this old Math SE thread which in my opinion also has far better answers than yours. Firstly, because (as pointed out by Arctic Char) your proof only works in the limited case that the two given functions are analytic, which is not only completely unnecessary but also mathematically irrelevant to the true essence of the theorem. Secondly, because the proof in the older thread very intuitively and clearly shows why the theorem is true, so much so that it is obvious to everyone that the usual combinatorial proof of the usual binomial theorem for natural exponents can be translated into a proof of the desired theorem.