The following is implied by the Tarski-Seidenberg Theorem :
A projection of a semi-algebraic set from dimension to dimension 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 leads to a semi-algebraic set in . 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.