Polynomial images of Rn

Let R be a real closed field and n 2. We prove that: (1) for every finite subset F of Rn, the semialgebraic set Rn F is a polynomial image of Rn; and (2) for any independent linear forms l1,…,lr of Rn, the semialgebraic set {l1>0,…,lr>0} Rn is a polynomial image of Rn.