∀ {M : Type u_1} [inst : CommMonoid M] {S : Submonoid M} {N : Type u_2} [inst_1 : CommMonoid N] {P : Type u_3}
  [inst_2 : CommMonoid P] (f : S.LocalizationMap N) {k : S.LocalizationMap P} {x : M},
  (f.ofMulEquivOfLocalizations (f.mulEquivOfLocalizations k)) x = k x