So there was this guy, named Stokes. And, in 1966, pick up sticks, he proved that it was actually better to leave the bottle of ketchup upside-down. Pretty sure he won the Noble Prize, plus American Idol for that discovery.
I had to look up "nerd sniping", I've been there. If it makes you feel any better, the Generalized Stokes' Theorem has a proof, e.g. it is a solved problem, it just requires a lot of reading.
I flipped through a few books in my e-library and found that Manifolds, Tensors, And Forms by Paul Renteln has two equivalent proofs starting on pg. 164. That was the "soonest" I could find the proof appearing in the books I know have a proof, e.g. building on the least material. IMO it's an "easy" book compared to other books I've read on manifolds and differential forms. There's a copy on LibGen.