AUSTRIAN-POLISH 1998 Problem 1

Let $x_1, x_2,y _1,y_2$ be real numbers such that $x_1^2 + x_2^2 \le 1$ . Prove the inequality $$(x_1y_1 + x_2y_2 - 1)^2 \ge (x_1^2 + x_2^2 - 1)(y_1^2 + y_2^2 -1)$$