-- Lean 4 proof step from Mathlib.FieldTheory.Galois.Profinite (goal)
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra k K
L : FiniteGaloisIntermediateField k K
g :
  (↑((limit (J := (FiniteGaloisIntermediateField k K)ᵒᵖ) (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop (P :=
        fun (X : TopCat.{u_4}) ↦ TotallyDisconnectedSpace.{u_4} (↑X : Type u_4))) :
    Type u_4)
x : ↥L.toIntermediateField
L' : FiniteGaloisIntermediateField k K
h : L ≤ L'
⊢ Eq.{u_4 + 1} (α := K)
    (↑((((proj L :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥L.toIntermediateField/k))
                g :
              Gal(↥L.toIntermediateField/k)) :
            (a : ↥L.toIntermediateField) → ↥L.toIntermediateField)
          x) :
      K)
    (↑((((proj L' :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥L'.toIntermediateField/k))
                g :
              Gal(↥L'.toIntermediateField/k)) :
            (a : ↥L'.toIntermediateField) → ↥L'.toIntermediateField)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L'.toIntermediateField) (↑x : K) ⋯)) :
      K)
have mk : ∀ (L : IntermediateField k K) [finiteDimensional : FiniteDimensional.{u_3, u_4} k ↥L]
  [isGalois : IsGalois.{u_3, u_4} k ↥L]
  (x :
    ↥{ toIntermediateField := L, finiteDimensional := finiteDimensional, isGalois := isGalois :
          FiniteGaloisIntermediateField k K }.toIntermediateField)
  (h :
    { toIntermediateField := L, finiteDimensional := finiteDimensional, isGalois := isGalois :
        FiniteGaloisIntermediateField k K } ≤
      L'),
  Eq.{u_4 + 1} (α := K)
    (↑((((proj
                    { toIntermediateField := L, finiteDimensional := finiteDimensional, isGalois := isGalois :
                      FiniteGaloisIntermediateField k K } :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥{ toIntermediateField := L, finiteDimensional := finiteDimensional,
                            isGalois := isGalois }.toIntermediateField/k))
                g :
              Gal(↥{ toIntermediateField := L, finiteDimensional := finiteDimensional, isGalois := isGalois :
                      FiniteGaloisIntermediateField k K }.toIntermediateField/k)) :
            (a :
                ↥{ toIntermediateField := L, finiteDimensional := finiteDimensional,
                      isGalois := isGalois }.toIntermediateField) →
              ↥{ toIntermediateField := L, finiteDimensional := finiteDimensional,
                    isGalois := isGalois }.toIntermediateField)
          x) :
      K)
    (↑((((proj L' :
                  (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                    Gal(↥L'.toIntermediateField/k))
                g :
              Gal(↥L'.toIntermediateField/k)) :
            (a : ↥L'.toIntermediateField) → ↥L'.toIntermediateField)
          (Subtype.mk (p := fun (x : K) ↦ x ∈ L'.toIntermediateField) (↑x : K) sorry)) :
      K)
Tactic: induction L with | _ L => ?_
Cleared hypotheses: L, x, h