∀ {α β : BoolAlg} (e : ↑α ≃o ↑β) (a b : ↑β),
  { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }.toFun (a ⊓ b) =
    { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }.toFun a ⊓
      { toFun := ⇑e.symm, map_sup' := ⋯, map_inf' := ⋯ }.toFun b