-- Module: Mathlib.FieldTheory.Galois.Profinite
Goal:
k : Type u_3
K : Type u_4
inst : Field k
inst_1 : Field K
inst_2 : Algebra 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)
L' : FiniteGaloisIntermediateField k K
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'
mk :
  ∀ (L' : IntermediateField k K) [finiteDimensional_1 : FiniteDimensional.{u_3, u_4} k ↥L']
    [isGalois_1 : IsGalois.{u_3, u_4} k ↥L']
    (h :
      { toIntermediateField := L, finiteDimensional := finiteDimensional, isGalois := isGalois :
          FiniteGaloisIntermediateField k K } ≤
        { toIntermediateField := L', finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 }),
    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
                      { toIntermediateField := L', finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 :
                        FiniteGaloisIntermediateField k K } :
                    (a : (↑(limit (asProfiniteGaloisGroupFunctor k K)).toProfinite.toTop : Type u_4)) →
                      Gal(↥{ toIntermediateField := L', finiteDimensional := finiteDimensional_1,
                              isGalois := isGalois_1 }.toIntermediateField/k))
                  g :
                Gal(↥{ toIntermediateField := L', finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 :
                        FiniteGaloisIntermediateField k K }.toIntermediateField/k)) :
              (a :
                  ↥{ toIntermediateField := L', finiteDimensional := finiteDimensional_1,
                        isGalois := isGalois_1 }.toIntermediateField) →
                ↥{ toIntermediateField := L', finiteDimensional := finiteDimensional_1,
                      isGalois := isGalois_1 }.toIntermediateField)
            (Subtype.mk (p := fun (x : K) ↦
              x ∈
                { toIntermediateField := L', finiteDimensional := finiteDimensional_1, isGalois := isGalois_1 :
                    FiniteGaloisIntermediateField k K }.toIntermediateField)
              (↑x : K) ⋯)) :
        K)
⊢ 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) ⋯)) :
      K)

Tactic:
induction L' with | _ L' => ?_