This is a joint work with Krzysztof Kurdyka and Adam Parusinski. We say a map f : Rn,0 → Rp is blow-analytic ([2]) if there is a composition σ : M → Rn of locally finitely many blow-ups so that f ◦σ is analytic. We say a map f : Rn,0 → Rp is arc-analytic ([3]) if f◦α is analytic for any analytic map α : R,0 → Rn,0. A blow-analytic map is clearly arc-analytic. It is known that ([1]) a semi- algebraic, arc analytic map is blow-analytic. If a bi-Lipschitz subanalytic homeomorphism is arc-analytic, then the inverse is arc-analytic ([4]). Let us consider a semi-algebraic homeomorphism h : Rn,0 → Rn,0. We show the following conditions are equivalent. • h is arc-analytic and h−1 is Lipschitz. • h−1 is arc-analytic and h is Lipschitz. The key step is to show that det(dh) is bounded away from infinity and zero. To show this, we need (at this moment at least) to compare virtual Poincare polynomials (or motivic measures) of partitions of arc space L(Rn, 0) with respect to certain Nash modification which sends everything normal crossing. In the talk, we describe the detailed proof of the following easier version: A (bi-)blow-analytic homeomotphism is bi-Lipschitz if it is Lipschitz and semi-algebraic.