A condition sufficient for nonexistence of a cycle in a two-dimensional system quadratic in one of the variables

For the system $\dot x=h_1(x)+h_2(x)y=P(x,y)$, $\dot y=f_1(x)+f_2(x)y+f_3(x)y^2=Q(x,y)$, the following theorem is proved.
Theorem. If the divergence of the vector field $(P,Q)$ does not change its sign and is not equal identically to zero along the isocline $h_1(x)+h_2(x)y=0$, then the system has no closed trajectory.