That is, if $A\subset{\mathbb{R}}^{n}\times{\mathbb{R}}^{m}$ is a semialgebraic set, and if $\pi$ is the projection onto the first $n$ coordinates, then $\pi(A)$ is also semialgebraic.

Łojasiewicz generalized this theorem further. For this we need a bit of notation.

Let $U\subset{\mathbb{R}}^{n}$. Suppose $\mathcal{A}(U)$ is any ring of real valued functions on $U$. Define $\mathcal{S}(\mathcal{A}(U))$ to be the smallest set of subsets of $U$, which contain the sets $\{x\in U\mid f(x)>0\}$ for all $f\in\mathcal{A}(U)$, and is closed under finite union, finite intersection and complement. Let $\mathcal{A}(U)[t]$ denote the ring of polynomials in $t\in{\mathbb{R}}^{m}$ with coefficients in $\mathcal{A}(U)$.