(1) Solvability of Equations P(f)=Q(g), where f, g meromorphic, P, Q rational functions over K; our result also contains the "Diophantic Equation" f^n+g^m=1, n>2, m>1, f, g entire which has no non-constant solutions. Interestingly, f^2+g^2=1 has no non constant entire solutions as well (this in particular shows that the sine and cosine are not entire!). This follows elementary.
(2) Uniqueness of meromorphic functions, that is, given a set S in K, f, g meromorphic maps. If the preimages of S under f resp. g coincide, one may conclude that f=g, supposed S is a set with sufficiently many points.