Let $a_1,b_1,c_1,a_2,b_2,c_2$ be positive real numbers. And $F,G:(0,\infty)\times (0, \infty) \to(0,\infty)$, differentiable and positive functions that satisfies: $$\frac{x}{F} = 1 + a_1x+ b_1y + c_1G$$
$$\frac{y}{G} = 1 + a_2x+ b_2y + c_2F.$$ Prove that if $0 < x_1 \le x_2$, and $0< y_2 \le y_1 $, then $F(x_1, y_1) \le F(x_2, y_2)$, and $G(x_1, y_1) \ge G(x_2, y_2)$.