Finsler's theorem asserts the equivalence of (i) and (ii) for pairs of real quadratic forms f and g on R-n: (i) f (xi) > 0 for all xi not equal 0 with g (xi) = 0; (ii) f - lambda g > 0 for some lambda is an element of R. We prove two extensions: 1, We admit a vector-valued quadratic form g: R-n --> R-k, for which we show that (i) implies that f - lambda.g > 0 on an (n -k + 1)-dimensional subspace Y subset of R-n for some lambda is an element of R-k. 2. In the nonstrict version of Finsler's theorem for indefinite g we replace R-n by a real vector space X.