  We prove a general version of the "Stability Theorem": if $K$ is a valued
field such that the ramification theoretical defect is trivial for all of its
finite extensions, and if $F|K$ is a finitely generated (transcendental)
extension of valued fields for which equality holds in the Abhyankar
inequality, then the defect is also trivial for all finite extensions of $F$.
This theorem is applied to eliminate ramification in such valued function
fields. It has applications to local uniformization and to the model theory of
valued fields in positive characteristic.
