∀ {𝕜 : Type u_1} {E : Type u_2} {G : Type u_4} [inst : CommRing 𝕜] [inst_1 : LinearOrder 𝕜] [IsStrictOrderedRing 𝕜]
  [inst_3 : CommRing E] [inst_4 : LinearOrder E] [IsStrictOrderedRing E] [inst_6 : AddCommGroup G] [inst_7 : Module 𝕜 G]
  [inst_8 : Module 𝕜 E] {s : Set G} [IsOrderedModule 𝕜 E] [IsScalarTower 𝕜 E E] [SMulCommClass 𝕜 E E] {f g : G → E},
  ConvexOn 𝕜 s f →
    ConcaveOn 𝕜 s g →
      (∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x) → (∀ ⦃x : G⦄, x ∈ s → g x ≤ 0) → AntivaryOn f g s → ConcaveOn 𝕜 s (f * g)