∀ {β : Type u_1} [inst : Lattice β] {P : β → Prop} (Psup : ∀ ⦃x y : β⦄, P x → P y → P (x ⊔ y))
  (Pinf : ∀ ⦃x y : β⦄, P x → P y → P (x ⊓ y)) (a b : { x // P x }),
  (SupHom.subtypeVal Psup).toFun (a ⊓ b) = (SupHom.subtypeVal Psup).toFun a ⊓ (SupHom.subtypeVal Psup).toFun b