Packing Odd Paths

Let s, t be vertices of a graph G, and let each edge e have a "capacity" c(e) ∈ R+. We prove a conjecture of Cook and Sebő that for every k ∈ R+, the following two statements are equivalent: (i) there is a "fractional packing" of value k of the odd length s - t paths, so that no edge is used more than its capacity; (ii) for every subgraph H of G with s, t ∈ V(H) in which there is no odd s - t path, [formula] ∑ (c(e): e ∈ E(G) − E(H), and e is incident with v ≥ 2k.