Lie algebras and separable morphisms in pro-affine algebraic groups

Let K be an algebraically closed field of arbitrary characteristic, and let f : G --> H be a surjective morphism of connected pro-affine algebraic groups over K. We show that if f is bijective and separable, then f is an isomorphism of pro-affine algebraic groups. Moreover, f is separable if and only if (its differential) f(o) is surjective. Furthermore, if f is separable, then L (Ker f) - Ker f(o).