Two functional identities agree on the image

Prove that if $f: \mathbb{R} \to \mathbb{R}$ is a monotonous function satisfying $f(f(x)) = f( - f(x)) = f(x)^2$, then $f(x) = f(-x)$ for all $x\in f(\mathbb{R})$.

Source: InternLM Lean-Workbook, record lean_workbook_14812 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.