In this article, we provide a Coq mechanised, executable, formal semantics for a realistic fragment of SQL consisting of select [distinct] from where group by having queries with NULL values, functions, aggregates, quantifiers and nested potentially correlated sub-queries. Relying on the Coq extraction mechanism to Ocaml, we produce a Coq certified semantic analyser for a SQL compiler. We then relate this fragment to a Coq formalised (extended) relational algebra that enjoys a bag semantics hence recovering all well-known algebraic equivalences upon which are based most of compilation optimisations. By doing so, we provide the first formally mechanised proof of the equivalence of SQL and extended relational algebra.