{𝕜 : Type u_1} →
  {V : Type u_2} →
    {V₂ : Type u_5} →
      {P : Type u_10} →
        {P₂ : Type u_11} →
          [inst : NormedField 𝕜] →
            [inst_1 : SeminormedAddCommGroup V] →
              [inst_2 : NormedSpace 𝕜 V] →
                [inst_3 : PseudoMetricSpace P] →
                  [inst_4 : NormedAddTorsor V P] →
                    [inst_5 : SeminormedAddCommGroup V₂] →
                      [inst_6 : NormedSpace 𝕜 V₂] →
                        [inst_7 : PseudoMetricSpace P₂] →
                          [inst_8 : NormedAddTorsor V₂ P₂] →
                            (toFun : P → P₂) →
                              (invFun : P₂ → P) →
                                (left_inv : autoParam (Function.LeftInverse invFun toFun) Equiv.left_inv._autoParam) →
                                  (right_inv :
                                      autoParam (Function.RightInverse invFun toFun) Equiv.right_inv._autoParam) →
                                    (linear : V ≃ₗ[𝕜] V₂) →
                                      (∀ (p : P) (v : V),
                                          { toFun := toFun, invFun := invFun, left_inv := left_inv,
                                                right_inv := right_inv }
                                              (v +ᵥ p) =
                                            linear v +ᵥ
                                              { toFun := toFun, invFun := invFun, left_inv := left_inv,
                                                  right_inv := right_inv }
                                                p) →
                                        (∀ (x : V), ‖linear x‖ = ‖x‖) → P ≃ᵃⁱ[𝕜] P₂