-- Lean 4 proof step from Mathlib.FieldTheory.PrimitiveElement (goal)
Goal:
F : Type u_1
inst : Field F
inst_1 : Infinite F
E : Type u_2
inst_2 : Field E
α β : E
inst_3 : Algebra F E
inst_4 : Algebra.IsSeparable F E
hα : IsIntegral F α
hβ : IsIntegral F β
f : F[X] := minpoly F α
g : F[X] := minpoly F β
ιFE : F →+* E := algebraMap F E
ιEE' : E →+* (Polynomial.map ιFE g).SplittingField := algebraMap E (Polynomial.map ιFE g).SplittingField
c : F
hc :
  ∀ α' ∈ (Polynomial.map (ιEE'.comp ιFE) f).roots,
    ∀ β' ∈ (Polynomial.map (ιEE'.comp ιFE) g).roots,
      -HSub.hSub (β := (Polynomial.map ιFE g).SplittingField) α'
              ((ιEE' : (a : E) → (Polynomial.map ιFE g).SplittingField) α : (Polynomial.map ιFE g).SplittingField) /
          HSub.hSub (β := (Polynomial.map ιFE g).SplittingField) β'
            ((ιEE' : (a : E) → (Polynomial.map ιFE g).SplittingField) β : (Polynomial.map ιFE g).SplittingField) ≠
        (ιEE'.comp ιFE : (a : F) → (Polynomial.map ιFE g).SplittingField) c
γ : E := α + c • β
p : (↥F⟮γ⟯)[X] :=
  EuclideanDomain.gcd
    (comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
      (HSub.hSub (α := (↥F⟮γ⟯)[X]) ((C : (a : ↥F⟮γ⟯) → (↥F⟮γ⟯)[X]) (AdjoinSimple.gen F γ) : (↥F⟮γ⟯)[X])
        (HMul.hMul (α := (↥F⟮γ⟯)[X]) (β := (↥F⟮γ⟯)[X])
          ((C : (a : ↥F⟮γ⟯) → (↥F⟮γ⟯)[X])
              (Subtype.mk (p := fun (x : E) ↦ Membership.mem (γ := IntermediateField F E) F⟮γ⟯ x)
                ((algebraMap F E : (a : F) → E) c) ⋯) :
            (↥F⟮γ⟯)[X])
          X)))
    (Polynomial.map (algebraMap F ↥F⟮γ⟯) g)
h : E[X] :=
  EuclideanDomain.gcd
    ((Polynomial.map ιFE f).comp
      (HSub.hSub (α := E[X]) ((C : (a : E) → E[X]) γ : E[X])
        (HMul.hMul (α := E[X]) (β := E[X]) ((C : (a : E) → E[X]) ((ιFE : (a : F) → E) c) : E[X]) X)))
    (Polynomial.map ιFE g)
map_g_ne_zero : Polynomial.map ιFE g ≠ 0
h_ne_zero : h ≠ 0
h_sep : h.Separable
h_root : eval β h = 0
h_splits : (Polynomial.map ιEE' h).Splits
h_roots : ∀ x ∈ (Polynomial.map ιEE' h).roots, x = (ιEE' : (a : E) → (Polynomial.map ιFE g).SplittingField) β
trans :
  Eq (α := E[X]) (Polynomial.map (algebraMap (↥F⟮γ⟯) E) p)
    (EuclideanDomain.gcd
      (Polynomial.map (algebraMap (↥F⟮γ⟯) E)
        (comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
          (HSub.hSub (α := (↥F⟮γ⟯)[X]) ((C : (a : ↥F⟮γ⟯) → (↥F⟮γ⟯)[X]) (AdjoinSimple.gen F γ) : (↥F⟮γ⟯)[X])
            (HMul.hMul (α := (↥F⟮γ⟯)[X]) (β := (↥F⟮γ⟯)[X])
              ((C : (a : ↥F⟮γ⟯) → (↥F⟮γ⟯)[X])
                  (Subtype.mk (p := fun (x : E) ↦ Membership.mem (γ := IntermediateField F E) F⟮γ⟯ x)
                    ((algebraMap F E : (a : F) → E) c) ⋯) :
                (↥F⟮γ⟯)[X])
              X))))
      (Polynomial.map (R := ↥F⟮γ⟯) (algebraMap (↥F⟮γ⟯) E) (Polynomial.map (algebraMap F ↥F⟮γ⟯) g)))
⊢ Polynomial.map (algebraMap (↥F⟮γ⟯) E) p = h
have trans₁ : EuclideanDomain.gcd
    (Polynomial.map (algebraMap (↥F⟮γ⟯) E)
      (comp (Polynomial.map (algebraMap F ↥F⟮γ⟯) f)
        (HSub.hSub (α := (↥F⟮γ⟯)[X]) ((C : (a : ↥F⟮γ⟯) → (↥F⟮γ⟯)[X]) (AdjoinSimple.gen F γ) : (↥F⟮γ⟯)[X])
          (HMul.hMul (α := (↥F⟮γ⟯)[X]) (β := (↥F⟮γ⟯)[X])
            ((C : (a : ↥F⟮γ⟯) → (↥F⟮γ⟯)[X])
                (Subtype.mk (p := fun (x : E) ↦ Membership.mem (γ := IntermediateField F E) F⟮γ⟯ x)
                  ((algebraMap F E : (a : F) → E) c) sorry) :
              (↥F⟮γ⟯)[X])
            X))))
    (Polynomial.map (R := ↥F⟮γ⟯) (algebraMap (↥F⟮γ⟯) E) (Polynomial.map (algebraMap F ↥F⟮γ⟯) g)) =
  h
Tactic: trans EuclideanDomain.gcd (?_ : E[X]) (?_ : E[X])