On an existence lemma in valuation theory

Recently one of the writers used, in proving a theorem on the commutativity of certain division rings, the following lemma :I. Let L be a field and K be its proper subfield. Except either when L is of characteristic p ≠ O and absolutely algebraic or ivhen L is algebraic and purely inseparable over K, there exists a pair of distinct (special exponential) valuations in L which coincide on K.II. Let K be a field which is either of characteristic 0 or not absolutely algebraic, and L be its separable finite extension. There exist then infinitely many valutions in L which are of 1st degree over K.