$R$ in each of the polynomials in the 2-dimensional projection to again get univariate polynomials. Find the roots of these polynomials and create two dimensional regions that form the stack over the region $R$. Designate sample points for these regions as well. Suppose we are given a CAD for $k$-dimensional real space, we use the projected set for $k+1$-dimensions and substitute the sample points of the $k$-dimensional CAD in them to get univariate polynomials. These can now be solved to get $k+1$-dimensional CAD.

Now, this information can be used by Algorithm 1 to perform quantifier elimination [@andreas]. Assume that the variables $x_1,...,x_k$ are quantifier-free and $x_{k+1},...,x_n$ are quantified, and let $Q_i$ be the corresponding quantifier for $i\in\{k+1,...,n\}$.

1.  For $k \leq i < n$, we have

    1.  If $Q_{i+1}$ is $\exists$, then a cell,$C \in D_{i}$ is valid if at least one cell in the stack over $C$ is valid.

    2.  If $Q_{i+1}$ is $\forall$, then a cell,$C \in D_{i}$ is valid if all cells in the stack over $C$ are valid.

2.  A region, $C$ in $D_{n}$ is valid if $\psi(t_{C})$ is TRUE, where $t_{C}$ is the sample point of that cell.

3.  Obtain the cells of $D_{k}$ which 