∀ {ιa : Type u_1} {ιb : Type u_2} [inst : Fintype ιa] [inst_1 : Fintype ιb] {R' : Type u_6} {Mᵢ : Type u_3}
  {N₁ : Type u_4} {N₂ : Type u_5} [inst_2 : CommSemiring R'] [inst_3 : AddCommGroup N₁] [inst_4 : Module R' N₁]
  [inst_5 : AddCommGroup N₂] [inst_6 : Module R' N₂] [inst_7 : AddCommMonoid Mᵢ] [inst_8 : Module R' Mᵢ]
  [inst_9 : DecidableEq ιa] [inst_10 : DecidableEq ιb] (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁) (b : Mᵢ [⋀^ιb]→ₗ[R'] N₂)
  (σ₁ σ₂ : Equiv.Perm (ιa ⊕ ιb)),
  (QuotientGroup.leftRel (Equiv.Perm.sumCongrHom ιa ιb).range) σ₁ σ₂ →
    (fun σ => Equiv.Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) σ₁ =
      (fun σ => Equiv.Perm.sign σ • MultilinearMap.domDomCongr σ ((↑a).domCoprod ↑b)) σ₂