The reference needed is page 263, Theorem 5.2.1 (dimension n). Please restate this theorem for dimension 2, in the autonomous case only. Do not include a proof, as the proof is a semester project already. Freely reference the theorem for dimension 2, in order to give the details of proof for the crossing statement. The crossing theorem is about the phase plot in xy. Give a complete statement of the result. Prove the result indirectly: suppose not, then obtain a contradiction. The statement contradicted is this one: P: The set of points on solution curve I is different than the set of points on solution curve II. You will end up showing, via Picard, that (not)P is true, which is a contradiction to the assumption that P is true. In the proof, you will need this lemma, which is easily proved for dimension n differential equations: LEMMA. For an autonomous differential equation X'(t)=f(X(t)), given a solution X(t) and a constant c, then X(t+c) is also a solution.