This article presents a weak system of intuitionistic second-order arithmetic, WKV, a subsystem of the one in S.C. Kleene, R.E. Vesley [The Foundations of Intuitionistic Mathematics: Especially in Relation to Recursive Functions, North-Holland Publishing Company, Amsterdam, 1965]. It is then shown that some statements of real analysis, like a version of the Heine–Borel Theorem, and some statements of logic, e.g. compactness of classical proposition calculus, are equivalent to the Fan Theorem in this system.