-- Module: Mathlib.Combinatorics.Configuration
Goal:
P : Type u_1
L : Type u_2
inst : Membership P L
inst_1 : HasLines P L
inst_2 : Fintype P
inst_3 : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
obtain :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    lineCount L p = pointCount P l
obtain₁ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    lineCount L p = pointCount P l
obtain₂ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l))
    (step1 : Eq (α := ℕ) (∑ i : P × L, lineCount L i.1) (∑ i : P × L, pointCount P i.2)), lineCount L p = pointCount P l
obtain₃ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    let s : Finset (P × L) := Set.toFinset {i : P × L | i.1 ∈ i.2};
    ∀ (step1 : Eq (α := ℕ) (∑ i : P × L, lineCount L i.1) (∑ i : P × L, pointCount P i.2))
      (step2 : Eq (α := ℕ) (∑ i ∈ s, lineCount L i.1) (∑ i ∈ s, pointCount P i.2)), lineCount L p = pointCount P l
obtain₄ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    let s : Finset (P × L) := Set.toFinset {i : P × L | i.1 ∈ i.2};
    ∀ (step1 : Eq (α := ℕ) (∑ i : P × L, lineCount L i.1) (∑ i : P × L, pointCount P i.2))
      (step2 : Eq (α := ℕ) (∑ i ∈ s, lineCount L i.1) (∑ i ∈ s, pointCount P i.2))
      (step3 : Eq (α := ℕ) (∑ i ∈ sᶜ, lineCount L i.1) (∑ i ∈ sᶜ, pointCount P i.2)), lineCount L p = pointCount P l
obtain₅ :
  ∀ (f : L → P) (hf1 : Function.Bijective f) (hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)),
    let s : Finset (P × L) := Set.toFinset {i : P × L | i.1 ∈ i.2};
    ∀ (step1 : Eq (α := ℕ) (∑ i : P × L, lineCount L i.1) (∑ i : P × L, pointCount P i.2))
      (step2 : Eq (α := ℕ) (∑ i ∈ s, lineCount L i.1) (∑ i ∈ s, pointCount P i.2))
      (step3 :
        Eq (α := ℕ) (∑ i ∈ Set.toFinset {i : P × L | i.1 ∈ i.2}ᶜ, lineCount L i.1)
          (∑ i ∈ Set.toFinset {i : P × L | i.1 ∈ i.2}ᶜ, pointCount P i.2)),
      lineCount L p = pointCount P l
⊢ lineCount L p = pointCount P l

Tactic:
classical
    obtain ⟨f, hf1, hf2⟩ := HasLines.exists_bijective_of_card_eq hPL
    let s : Finset (P × L) := Set.toFinset { i | i.1 ∈ i.2 }
    have step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2 := by
      rw [← Finset.univ_product_univ, Finset.sum_product_right, Finset.sum_product]
      simp_rw [Finset.sum_const, Finset.card_univ, hPL, sum_lineCount_eq_sum_pointCount]
    have step2 : ∑ i ∈ s, lineCount L i.1 = ∑ i ∈ s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      on_goal 1 =>
        rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }, eq_comm]
        · refine sum_bijective _ hf1 (by simp) fun l _ ↦ ?_
          simp_rw [hf2, sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
          change pointCount P l • _ = lineCount L (f l) • _
          rw [hf2]
      all_goals simp_rw [s, Finset.mem_univ, true_and, Set.mem_toFinset]; exact fun p => Iff.rfl
    have step3 : ∑ i ∈ sᶜ, lineCount L i.1 = ∑ i ∈ sᶜ, pointCount P i.2 := by
      rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1
    rw [← Set.toFinset_compl] at step3
    exact
      ((Finset.sum_eq_sum_iff_of_le fun i hi =>
              HasLines.pointCount_le_lineCount (by exact Set.mem_toFinset.mp hi)).mp
          step3.symm (p, l) (Set.mem_toFinset.mpr hpl)).symm

Premises used: Set.toFinset_compl, Configuration.HasLines.exists_bijective_of_card_eq