Then one of the following statements holds: (A) there exists ¯ x ∈ X such that g (¯ x, y ) ≥ λ, for all y ∈ Y ; or (B) there exists ¯ y ∈ Y such that f ( x, ¯ y ) ≤ λ, for all x ∈ X. Proof. Simply apply Theorem 12 to A, B ⊂ X × Y defined as: Note that in view of (i) a coincidence between A and B is impossible as it yields λ < λ. Since all hypotheses of Theorem 12 are satisfied save for A ( x ) =  for all x ∈ X and B − $^{1}$( y ) =  for all y ∈ Y, it follows that either A (¯ x ) =  for some ¯ x ∈ X (thesis (A)) or B − $^{1}$(¯ y ) =  for some ¯ y ∈ Y (thesis (B)). Remark 18 Theorem 17 implies α = sup$_{X}$ inf$_{Y}$ g ( x, y ) ≥ inf$_{Y}$ sup$_{X}$ f ( x, y ) = β. Indeed, assuming that α < β < ∞ , let λ be an arbitrary but fixed real number strictly between α and β. By Theorem 17, either there exists ¯ y ∈ Y such that f ( x, ¯ y ) ≤ λ, for all x ∈ X thus β ≤ λ < β which is impossible, or there exists ¯ x ∈ X such that g (¯ x, y ) ≥ λ, for all y ∈ Y thus α ≥ λ > α which is absurd. Hence α ≥ β. Maurice Sion’s formulation of the von Neumann Minimax Theorem follows immediately with f = g (see [1]): Corollary 19 (Sion-von Neumann Minimax Theorem) Let X and Y be convex subsets of topological vector spaces and let f be a real function on X × Y such that: (i) x ↦→ f ( x, y ) is quasiconcave and u.s.c. on X for each fixed y ∈ Y ; (ii) y ↦→ f ( x, y ) is quasiconvex and l.s.c. on Y for each fixed x ∈ X ; A := { ( x, y ) : g ( x, y ) < λ } and B := { ( x, y ) : f ( x, y ) > λ } . 17