theorem IsPushout.isVanKampen_inl {W E X Z : C} (c : BinaryCofan W E) [FinitaryExtensive C]
    [HasPullbacks C] (hc : IsColimit c) (f : W ⟶ X) (h : X ⟶ Z) (i : c.pt ⟶ Z)
    (H : IsPushout f c.inl h i) : H.IsVanKampen := by
  obtain ⟨hc₁⟩ := (is_coprod_iff_isPushout c hc H.1).mpr H
  introv W' hf hg hh hi w
  obtain ⟨hc₂⟩ := ((BinaryCofan.isVanKampen_iff _).mp (FinitaryExtensive.vanKampen c hc)
    (BinaryCofan.mk _ pullback.fst) _ _ _ hg.w.symm pullback.condition.symm).mpr
    ⟨hg, IsPullback.of_hasPullback αY c.inr⟩
  refine' (is_coprod_iff_isPushout _ hc₂ w).symm.trans _
  refine' ((BinaryCofan.isVanKampen_iff _).mp (FinitaryExtensive.vanKampen _ hc₁)
    (BinaryCofan.mk _ _) pullback.snd _ _ _ hh.w.symm).trans _
  · dsimp; rw [← pullback.condition_assoc, Category.assoc, hi.w]
  constructor
  · rintro ⟨hc₃, hc₄⟩
    refine' ⟨hc₄, _⟩
    let Y'' := pullback αZ i
    let cmp : Y' ⟶ Y'' := pullback.lift i' αY hi.w
    have e₁ : (g' ≫ cmp) ≫ pullback.snd = αW ≫ c.inl := by
      rw [Category.assoc, pullback.lift_snd, hg.w]
    have e₂ : (pullback.fst ≫ cmp : pullback αY c.inr ⟶ _) ≫ pullback.snd = pullback.snd ≫ c.inr :=
      by rw [Category.assoc, pullback.lift_snd, pullback.condition]
    obtain ⟨hc₄⟩ := ((BinaryCofan.isVanKampen_iff _).mp (FinitaryExtensive.vanKampen c hc)
      (BinaryCofan.mk _ _) αW _ _ e₁.symm e₂.symm).mpr <| by
        constructor
        · apply IsPullback.of_right _ e₁ (IsPullback.of_hasPullback _ _)
          rw [Category.assoc, pullback.lift_fst, ← H.w, ← w.w]; exact hf.paste_horiz hc₄
        · apply IsPullback.of_right _ e₂ (IsPullback.of_hasPullback _ _)
          rw [Category.assoc, pullback.lift_fst]; exact hc₃
    · 

Proof state:
case intro.intro.refine'_2.mp.intro.intro
J : Type v'
inst✝³ : Category.{u', v'} J
C : Type u
inst✝² : Category.{v, u} C
W✝ X✝ Y Z✝ : C
f✝ : W✝ ⟶ X✝
g : W✝ ⟶ Y
h✝ : X✝ ⟶ Z✝
i✝ : Y ⟶ Z✝
W E X Z : C
c : BinaryCofan W E
inst✝¹ : FinitaryExtensive C
inst✝ : HasPullbacks C
hc : IsColimit c
f : W ⟶ X
h : X ⟶ Z
i : c.pt ⟶ Z
H : IsPushout f (BinaryCofan.inl c) h i
hc₁ : IsColimit (BinaryCofan.mk (BinaryCofan.inr c ≫ i) h)
W' X' Y' Z' : C
f' : W' ⟶ X'
g' : W' ⟶ Y'
h' : X' ⟶ Z'
i' : Y' ⟶ Z'
αW : W' ⟶ W
αX : X' ⟶ X
αY : Y' ⟶ ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := WalkingPair.left }
αZ : Z' ⟶ Z
hf : IsPullback f' αW αX f
hg : IsPullback g' αW αY (BinaryCofan.inl c)
hh : CommSq h' αX αZ h
hi : CommSq i' αY αZ i
w : CommSq f' g' h' i'
hc₂ : IsColimit (BinaryCofan.mk g' pullback.fst)
hc₃ :
  IsPullback (BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr (BinaryCofan.mk g' pullback.fst) ≫ i') h')) pullback.snd
    αZ (BinaryCofan.inl (BinaryCofan.mk (BinaryCofan.inr c ≫ i) h))
hc₄✝ :
  IsPullback (BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr (BinaryCofan.mk g' pullback.fst) ≫ i') h')) αX αZ
    (BinaryCofan.inr (BinaryCofan.mk (BinaryCofan.inr c ≫ i) h))
Y'' : C := pullback αZ i
cmp : Y' ⟶ Y'' := pullback.lift i' αY (_ : i' ≫ αZ = αY ≫ i)
e₁ : (g' ≫ cmp) ≫ pullback.snd = αW ≫ BinaryCofan.inl c
e₂ : (pullback.fst ≫ cmp) ≫ pullback.snd = pullback.snd ≫ BinaryCofan.inr c
hc₄ : IsColimit (BinaryCofan.mk (g' ≫ cmp) (pullback.fst ≫ cmp))
⊢ IsPullback i' αY αZ i

Next tactic:
rw [← Category.id_comp αZ, ← show cmp ≫ pullback.snd = αY from pullback.lift_snd _ _ _]