Two shifted square-root squared reciprocals have a sum lower bound

Prove that for $x,y\in\mathbb{R^ + }$ , $\frac {1}{(1 + \sqrt {x})^{2}} + \frac {1}{(1 + \sqrt {y})^{2}} \ge \frac {2}{x + y + 2}$

Source: InternLM Lean-Workbook, record lean_workbook_19966 (Apache-2.0). Complete source proposition preserved; proof developed independently.