-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Basic (goal)
Goal:
σ : Type u_1
A : Type u
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
X : Scheme
f : A →+* (↑Γ(X, ⊤) : Type u)
hf : Eq (α := Ideal (↑Γ(X, ⊤) : Type u)) (Ideal.map f (HomogeneousIdeal.toIdeal (HomogeneousIdeal.irrelevant 𝒜))) ⊤
x : (openCoverOfMapIrrelevantEqTop 𝒜 f hf).I₀
⊢ Eq (α :=
    Spec
        (CommRingCat.of
          (Localization.Away (M := (↑Γ(X, ⊤) : Type u))
            ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
                (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                  0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                    r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))) :
              (↑Γ(X, ⊤) : Type u)))) ⟶
      Spec (CommRingCat.of ↥(𝒜 0)))
    (Spec.map
      (CommRingCat.ofHom
        (@RingHom.comp _
          (HomogeneousLocalization 𝒜
            (Submonoid.powers
              (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                  r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i))))))
          _ _ Semiring.toNonAssocSemiring _
          (@RingHom.comp _
            (Localization.Away
              (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                  r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))
            _ _ Semiring.toNonAssocSemiring _
            (IsLocalization.map
              (Localization.Away (M := (↑Γ(X, ⊤) : Type u))
                ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
                    (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                      0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                        r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))) :
                  (↑Γ(X, ⊤) : Type u)))
              f ⋯)
            (algebraMap
              (Away 𝒜
                (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                  0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                    r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))
              (Localization.Away
                (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                  0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                    r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))))
          (algebraMap (↥(𝒜 0))
            (HomogeneousLocalization 𝒜
              (Submonoid.powers
                (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                  0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                    r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i))))))))))
    (Spec.map
      (CommRingCat.ofHom
        (RingHom.comp
          (algebraMap (↑Γ(X, ⊤) : Type u)
            (Localization.Away (M := (↑Γ(X, ⊤) : Type u))
              ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
                  (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                    0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                      r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))) :
                (↑Γ(X, ⊤) : Type u))))
          (f.comp (algebraMap (↥(𝒜 0)) A)))))
have e_a : Eq (α :=
  ↥(𝒜 0) →+*
    Localization.Away (M := (↑Γ(X, ⊤) : Type u))
      ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
          (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
            0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
              r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))) :
        (↑Γ(X, ⊤) : Type u)))
  (@RingHom.comp _
    (HomogeneousLocalization 𝒜
      (Submonoid.powers
        (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
          0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
            r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i))))))
    _ _ Semiring.toNonAssocSemiring _
    (@RingHom.comp _
      (Localization.Away
        (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
          0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
            r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))
      _ _ Semiring.toNonAssocSemiring _
      (IsLocalization.map
        (Localization.Away (M := (↑Γ(X, ⊤) : Type u))
          ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
              (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
                0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                  r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))) :
            (↑Γ(X, ⊤) : Type u)))
        f sorry)
      (algebraMap
        (Away 𝒜
          (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
            0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
              r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))
        (Localization.Away
          (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
            0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
              r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))))))
    (algebraMap (↥(𝒜 0))
      (HomogeneousLocalization 𝒜
        (Submonoid.powers
          (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
            0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
              r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i))))))))
  (RingHom.comp
    (algebraMap (↑Γ(X, ⊤) : Type u)
      (Localization.Away (M := (↑Γ(X, ⊤) : Type u))
        ((f : (a : A) → (↑Γ(X, ⊤) : Type u))
            (x.snd (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i).fst (β := fun (r : A) ↦
              0 < x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i) ∧
                r ∈ 𝒜 (x.fst (β := fun (i : ℕ) ↦ (r : A) ×' 0 < i ∧ r ∈ 𝒜 i)))) :
          (↑Γ(X, ⊤) : Type u))))
    (f.comp (algebraMap (↥(𝒜 0)) A)))
Tactic: congr 2