Semi-algebraic sets and semi-algebraic functions are essential to specify and certify cylindrical algebraic decomposition algorithms. We formally define in Coq the base operations on semi-algebraic sets and functions using embedded first-order formulae over the language of real closed fields, and we prove the correctness of their geometrical interpretation. In doing so, we exploit a previous formalisation of quantifier elimination on such embedded formulae to guarantee the decidability of several first-order properties and keep our development constructive. We also exploit it to formalise formulae substitution without having to handle bound variables.