There are more uses of the concept: Transversal, given in Coddington-Levinson book of Theory of ODE?

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I'm reading the Poincaré-Bendixson Theorem's proof in the book I've mentioned. I feel kind of curious about the concept shown there: the transversal, and how they used it. I would like to know if there are other uses of this concept in other proofs, specially if they are oriented to study properties in ODE's. Thank you.