∀ {ι : Type u_1} {G : ι → Type u_2} {H : Type u_3} [inst : (i : ι) → Group (G i)] [inst_1 : Group H]
  {φ : (i : ι) → H →* G i} {d : Monoid.PushoutI.NormalWord.Transversal φ} [inst_2 : DecidableEq ι]
  [inst_3 : (i : ι) → DecidableEq (G i)] {i : ι} (g : G i) (w : Monoid.PushoutI.NormalWord d),
  g • w =
    (Monoid.PushoutI.NormalWord.equivPair i).symm
      (let __src := (Monoid.PushoutI.NormalWord.equivPair i) w;
      { head := g * ((Monoid.PushoutI.NormalWord.equivPair i) w).head, tail := __src.tail, fstIdx_ne := ⋯,
        normalized := ⋯ })