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This paper shows that the subtyping relation of a higher-order lambda calculus, F≤ω, is anti-symmetric. It exhibits the first such proof, establishing in the process that the subtyping relation is a partial order--reflexive, transitive, and anti-symmetric up to β-equality. While a subtyping relation is reflexive and transitive by definition, antisymmetry is a derived property. The result, which may seem obvious to the nonexpert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping offers a powerful new technology to solve the problem: of particular importance is our extended rule for the well-formedness of types with head variables. The paper also gives a presentation of F≤ω without a relation for β-equality, apparently the first such, and shows its equivalence with the traditional presentation.