-- Lean 4 proof step from Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Proper (goal)
Goal:
σ : Type u_1
A : Type u_2
inst : CommRing A
inst_1 : SetLike σ A
inst_2 : AddSubgroupClass σ A
𝒜 : ℕ → σ
inst_3 : GradedRing 𝒜
i j :
  PreZeroHypercover.I₀ (S := Proj 𝒜)
    (Precoverage.ZeroHypercover.toPreZeroHypercover (AffineOpenCover.openCover (affineOpenCover 𝒜)))
e₁ : pullback (AffineCover.f (P := IsOpenImmersion) (affineOpenCover 𝒜) i ≫ toSpecZero 𝒜)
    (AffineCover.f (P := IsOpenImmersion) (affineOpenCover 𝒜) j ≫ toSpecZero 𝒜) ≅
  Spec
    (CommRingCat.of
      (Away 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⊗[↥(𝒜 0)]
        Away 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A))) :=
  congrHom ⋯ ⋯ ≪≫
    pullbackSpecIso (↥(𝒜 0)) (Away 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A))
      (Away 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A))
e₂ : pullback (AffineCover.f (P := IsOpenImmersion) (affineOpenCover 𝒜) i)
    (AffineCover.f (P := IsOpenImmersion) (affineOpenCover 𝒜) j) ≅
  Spec
    (CommRingCat.of
      (Away 𝒜
        ((↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) * (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A)))) :=
  pullbackAwayιIso 𝒜 ⋯ ⋯ ⋯ ⋯ ⋯
⊢ IsClosedImmersion (X :=
    pullback (X := Spec (AffineCover.X (P := IsOpenImmersion) (affineOpenCover 𝒜) i))
      (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯)
      (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯))
    (Y :=
    pullback (X := Spec (AffineCover.X (P := IsOpenImmersion) (affineOpenCover 𝒜) i)) (Y :=
      Spec (AffineCover.X (P := IsOpenImmersion) (affineOpenCover 𝒜) j))
      (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜)
      (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜))
    (CategoryStruct.comp (Y :=
      pullback (X := Proj 𝒜) (Y :=
        pullback (X := Spec (AffineCover.X (P := IsOpenImmersion) (affineOpenCover 𝒜) i)) (Y :=
          Spec (AffineCover.X (P := IsOpenImmersion) (affineOpenCover 𝒜) j))
          (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜)
          (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜))
        (Z := diagonalObj (X := Proj 𝒜) (toSpecZero 𝒜)) (diagonal (toSpecZero 𝒜))
        (pullback.map (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜)
          (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜) (toSpecZero 𝒜) (toSpecZero 𝒜)
          (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯)
          (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯) (𝟙 (Spec (CommRingCat.of ↥(𝒜 0)))) ⋯ ⋯))
      (Iso.inv
        (pullbackDiagonalMapIdIso (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯)
          (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯) (toSpecZero 𝒜)))
      (snd (diagonal (toSpecZero 𝒜))
        (pullback.map (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜)
          (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯ ≫ toSpecZero 𝒜) (toSpecZero 𝒜) (toSpecZero 𝒜)
          (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯)
          (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) ⋯ ⋯) (𝟙 (Spec (CommRingCat.of ↥(𝒜 0)))) ⋯ ⋯)))
have rw : IsClosedImmersion
  (e₂.inv ≫
    CategoryStruct.comp (Y :=
        pullback (X := Proj 𝒜) (Y :=
          pullback (X := Spec (AffineCover.X (P := IsOpenImmersion) (affineOpenCover 𝒜) i)) (Y :=
            Spec (AffineCover.X (P := IsOpenImmersion) (affineOpenCover 𝒜) j))
            (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry ≫ toSpecZero 𝒜)
            (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry ≫ toSpecZero 𝒜))
          (Z := diagonalObj (X := Proj 𝒜) (toSpecZero 𝒜)) (diagonal (toSpecZero 𝒜))
          (pullback.map (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry ≫ toSpecZero 𝒜)
            (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry ≫ toSpecZero 𝒜) (toSpecZero 𝒜)
            (toSpecZero 𝒜) (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry)
            (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry) (𝟙 (Spec (CommRingCat.of ↥(𝒜 0)))) sorry sorry))
        (Iso.inv
          (pullbackDiagonalMapIdIso (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry)
            (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry) (toSpecZero 𝒜)))
        (snd (diagonal (toSpecZero 𝒜))
          (pullback.map (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry ≫ toSpecZero 𝒜)
            (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry ≫ toSpecZero 𝒜) (toSpecZero 𝒜)
            (toSpecZero 𝒜) (awayι 𝒜 (↑(i.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry)
            (awayι 𝒜 (↑(j.snd (β := fun (i : ℕ+) ↦ ↥(𝒜 (↑i : ℕ)))) : A) sorry sorry) (𝟙 (Spec (CommRingCat.of ↥(𝒜 0)))) sorry sorry)) ≫
      e₁.hom)
Tactic: rw [← MorphismProperty.cancel_right_of_respectsIso (P := @IsClosedImmersion) _ e₁.hom,
    ← MorphismProperty.cancel_left_of_respectsIso (P := @IsClosedImmersion) e₂.inv]