theorem affineLocally_stableUnderComposition : (affineLocally @P).StableUnderComposition := by
  intro X Y S f g hf hg
  let 𝒰 : ∀ i, ((S.affineCover.pullbackCover (f ≫ g)).obj i).OpenCover := by
    intro i
    refine' Scheme.OpenCover.bind _ fun i => Scheme.affineCover _
    apply Scheme.OpenCover.pushforwardIso _
      (pullbackRightPullbackFstIso g (S.affineCover.map i) f).hom
    apply Scheme.Pullback.openCoverOfRight
    exact (pullback g (S.affineCover.map i)).affineCover
  -- Porting note: used to be - rw [hP.affine_openCover_iff (f ≫ g) S.affineCover _] - but
  -- metavariables cause problems in the instance search
  apply (@affine_openCover_iff _ hP _ _ (f ≫ g) S.affineCover _ ?_ ?_).mpr
  

Proof state:
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : PropertyIsLocal P
X Y S : Scheme
f : X ⟶ Y
g : Y ⟶ S
hf : affineLocally P f
hg : affineLocally P g
𝒰 : (i : (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)).J) →
  Scheme.OpenCover (Scheme.OpenCover.obj (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)) i) :=
  fun i =>
    Scheme.OpenCover.bind
      (Scheme.OpenCover.pushforwardIso
        (Scheme.Pullback.openCoverOfRight
          (Scheme.affineCover (pullback g (Scheme.OpenCover.map (Scheme.affineCover S) i))) f pullback.fst)
        (pullbackRightPullbackFstIso g (Scheme.OpenCover.map (Scheme.affineCover S) i) f).hom)
      fun i_1 =>
      Scheme.affineCover
        (Scheme.OpenCover.obj
          (Scheme.OpenCover.pushforwardIso
            (Scheme.Pullback.openCoverOfRight
              (Scheme.affineCover (pullback g (Scheme.OpenCover.map (Scheme.affineCover S) i))) f pullback.fst)
            (pullbackRightPullbackFstIso g (Scheme.OpenCover.map (Scheme.affineCover S) i) f).hom)
          i_1)
⊢ ∀ (i : (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)).J) (j : (?m.242581 i).J),
    P (Scheme.Γ.map (Scheme.OpenCover.map (?m.242581 i) j ≫ pullback.snd).op)
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : PropertyIsLocal P
X Y S : Scheme
f : X ⟶ Y
g : Y ⟶ S
hf : affineLocally P f
hg : affineLocally P g
𝒰 : (i : (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)).J) →
  Scheme.OpenCover (Scheme.OpenCover.obj (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)) i) :=
  fun i =>
    Scheme.OpenCover.bind
      (Scheme.OpenCover.pushforwardIso
        (Scheme.Pullback.openCoverOfRight
          (Scheme.affineCover (pullback g (Scheme.OpenCover.map (Scheme.affineCover S) i))) f pullback.fst)
        (pullbackRightPullbackFstIso g (Scheme.OpenCover.map (Scheme.affineCover S) i) f).hom)
      fun i_1 =>
      Scheme.affineCover
        (Scheme.OpenCover.obj
          (Scheme.OpenCover.pushforwardIso
            (Scheme.Pullback.openCoverOfRight
              (Scheme.affineCover (pullback g (Scheme.OpenCover.map (Scheme.affineCover S) i))) f pullback.fst)
            (pullbackRightPullbackFstIso g (Scheme.OpenCover.map (Scheme.affineCover S) i) f).hom)
          i_1)
⊢ (i : (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)).J) →
    Scheme.OpenCover (Scheme.OpenCover.obj (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)) i)
P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop
hP : PropertyIsLocal P
X Y S : Scheme
f : X ⟶ Y
g : Y ⟶ S
hf : affineLocally P f
hg : affineLocally P g
𝒰 : (i : (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)).J) →
  Scheme.OpenCover (Scheme.OpenCover.obj (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)) i) :=
  fun i =>
    Scheme.OpenCover.bind
      (Scheme.OpenCover.pushforwardIso
        (Scheme.Pullback.openCoverOfRight
          (Scheme.affineCover (pullback g (Scheme.OpenCover.map (Scheme.affineCover S) i))) f pullback.fst)
        (pullbackRightPullbackFstIso g (Scheme.OpenCover.map (Scheme.affineCover S) i) f).hom)
      fun i_1 =>
      Scheme.affineCover
        (Scheme.OpenCover.obj
          (Scheme.OpenCover.pushforwardIso
            (Scheme.Pullback.openCoverOfRight
              (Scheme.affineCover (pullback g (Scheme.OpenCover.map (Scheme.affineCover S) i))) f pullback.fst)
            (pullbackRightPullbackFstIso g (Scheme.OpenCover.map (Scheme.affineCover S) i) f).hom)
          i_1)
⊢ ∀ (i : (Scheme.OpenCover.pullbackCover (Scheme.affineCover S) (f ≫ g)).J) (j : (?m.242581 i).J),
    IsAffine (Scheme.OpenCover.obj (?m.242581 i) j)

Next tactic:
rotate_left