Independence can be lost on a subinterval — but not for solutions of a linear ODE

Consider the statements: I: If $\phi_1,\phi_2$ are linearly independent functions on an interval I, they are linearly independent on any interval J contained inside I. II: If $\phi_1,\phi_2$ are linearly independent solutions of $L(y)=0$ on an interval I, they are linearly independent on any interval J contained inside I. Then: