Prove the following inequality $x^2+y^2+1>x\sqrt{y^2+1}+y\sqrt{x^2+1}$ Can anybody help me prove this inequality ?

Hint
For $x \ge 0$ and $y \ge 0$ use  
AM-GM
$$x^2+(y^2+1) \ge 2\sqrt{x^2(y^2+1)}=2x\sqrt{y^2+1}$$
$$(x^2+1)+y^2 \ge 2\sqrt{y^2(x^2+1)}=2y\sqrt{x^2+1}$$
Sum both and prove that the equality is not possible.
Also think about $x<0$ or $y<0$. That should be easy.
Can you finish?
