Semi-algebraic projections are semi-algebraic

The following is implied by the Tarski-Seidenberg Theorem [77]:

A projection of a semi-algebraic set from dimension $n$ to dimension $n-1$ is a semi-algebraic set.

This gives a kind of closure of semi-algebraic sets under projection, which is required to ensure that every projection of a semi-algebraic set in ${\mathbb{R}}^i$ leads to a semi-algebraic set in ${\mathbb{R}}^{i-1}$. This property is actually not true for (real) algebraic varieties, which were introduced in Section 4.4.1. Varieties are defined using only the $=$ relation and are not closed under the projection operation. Therefore, it is a good thing (not just a coincidence!) that we are using semi-algebraic sets.