/-- Given a continuous multilinear map `f` in `n+1` variables, split the first variable to obtain
a continuous linear map into continuous multilinear maps in `n` variables, given by
`x ↦ (m ↦ f (cons x m))`. -/
def ContinuousMultilinearMap.curryLeft (f : ContinuousMultilinearMap 𝕜 Ei G) :
    Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G :=
  LinearMap.mkContinuous
    { -- define a linear map into `n` continuous multilinear maps
      -- from an `n+1` continuous multilinear map
      toFun := fun x =>
        (f.toMultilinearMap.curryLeft x).mkContinuous (‖f‖ * ‖x‖) (f.norm_map_cons_le x)
      map_add' := fun x y => by
        ext m
        exact f.cons_add m x y
      map_smul' := fun c x => by
        ext m
        

Proof state:
case H
𝕜 : Type u
ι : Type v
ι' : Type v'
n : ℕ
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
Ei : Fin (Nat.succ n) → Type wEi
G : Type wG
G' : Type wG'
inst✝¹⁴ : Fintype ι
inst✝¹³ : Fintype ι'
inst✝¹² : NontriviallyNormedField 𝕜
inst✝¹¹ : (i : ι) → NormedAddCommGroup (E i)
inst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁹ : (i : ι) → NormedAddCommGroup (E₁ i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁷ : (i : ι') → NormedAddCommGroup (E' i)
inst✝⁶ : (i : ι') → NormedSpace 𝕜 (E' i)
inst✝⁵ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)
inst✝⁴ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : NormedAddCommGroup G'
inst✝ : NormedSpace 𝕜 G'
f : ContinuousMultilinearMap 𝕜 Ei G
c : 𝕜
x : Ei 0
m : (i : Fin n) → Ei (succ i)
⊢ (AddHom.toFun
        {
          toFun := fun x =>
            MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
              (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖),
          map_add' :=
            (_ :
              ∀ (x y : Ei 0),
                (fun x =>
                      MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
                        (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖))
                    (x + y) =
                  (fun x =>
                        MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
                          (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖))
                      x +
                    (fun x =>
                        MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
                          (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖))
                      y) }
        (c • x))
      m =
    ((RingHom.id 𝕜) c •
        AddHom.toFun
          {
            toFun := fun x =>
              MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
                (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖),
            map_add' :=
              (_ :
                ∀ (x y : Ei 0),
                  (fun x =>
                        MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
                          (_ : ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖))
                      (x + y) =
                    (fun x =>
                          MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
                            (_ :
                              ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖))
                        x +
                      (fun x =>
                          MultilinearMap.mkContinuous ((MultilinearMap.curryLeft f.toMultilinearMap) x) (‖f‖ * ‖x‖)
                            (_ :
                              ∀ (m : (i : Fin n) → Ei (succ i)), ‖f (Fin.cons x m)‖ ≤ ‖f‖ * ‖x‖ * ∏ i : Fin n, ‖m i‖))
                        y) }
          x)
      m

Next tactic:
exact
          f.cons_smul m c x