A quick proof of Seymour's theorem on t-joins

Let G be a graph and t : V ( G ) , {0, 1}, where t(V(G)) is even. (If X~_ V(G), then t ( X ) : = E { t (x):xeX}. ) A t-join is a set F ~ E ( G ) with d~(x)=t(x) (mod 2), Vx e V(G). (dF(x) denotes the number of edges of F incident with x, where loops count twice.) t-joins contain Chinese postman tours, matchings and minimum weight paths as a special case. (el. [1, 7]). If X c V(G), let 6 ( X ) = {xy e E(G): y qX, x eX} . If t (X)~ 1 (mod 2), then 6(X) is called a t-cut, t-cuts contain plane multicommodity flows as a special case [8]. For basic definitions concerning graphs we refer to [4]. Let ~(G, t)=min{lF: F ~E(G), F is a t-join}, and v(G, t) =max(ICl: C is a family of disjoint t-cuts}. It is easy to see that r(G, t) >I v(G, t).