Quasimonotone extension and initial value problems

We shall prove an extension lemma concerning quasimonotone increasing funct ions. Applying this result to (x) over dot = f(t,x),x(0) = x(0), where f is quasimonotone increasing, it is shown that two solutions x(1) and x(2) sat isfying x(1) less than or equal to x(2), x(1) not equal x(2) can be separat ed by a solution y different from x(1) and x(2) : x(1) less than or equal t o y less than or equal to x(2), y not equal x(1), x(2).