This is really an answer by Yemon Choi (who put this in a comment, which is now buried) but as it's community wiki...
A very cute proof of just the fact that an bases have the same (finite) cardinality is given by Ford in http://www.ams.org/mathscinet-getitem?mr=1328020
Sadly it doesn't work in characteristic p, which my course is meant to touch upon.
