IR-MO 2004 Problem 10

$f:\mathbb{R}^2 \to \mathbb{R}^2$ is injective and surjective. Distance of $X$ and $Y$ is not less than distance of $f(X)$ and $f(Y)$ . Prove for $A$ in plane: $$S(A) \geq S(f(A))$$ where $S(A)$ is area of $A$