l3lab/ntp-mathlib huggingface dataset:
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; 

Proof state:
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ {i | i.1 ∈ i.2} ↔ p.1 ∈ {p_1 | p_1 ∈ p.2}

Next tactic:
exact fun p => Iff.rfl
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals 

Proof state:
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ s ↔ p.1 ∈ Finset.univ ∧ p.2 ∈ Set.toFinset {l | p.1 ∈ l}

Next tactic:
simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; 

Proof state:
case h
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
⊢ ∀ (p : P × L), p ∈ {i | i.1 ∈ i.2} ↔ p.2 ∈ {l | p.1 ∈ l}

Next tactic:
exact fun p => Iff.rfl
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
    

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2
⊢ lineCount L p = pointCount P l

Next tactic:
have step3 : ∑ i in sᶜ, lineCount L i.1 = ∑ i in sᶜ, pointCount P i.2 := by
      rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
    have step3 : ∑ i in sᶜ, lineCount L i.1 = ∑ i in sᶜ, pointCount P i.2 := by
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2
⊢ ∑ i in sᶜ, lineCount L i.1 = ∑ i in sᶜ, pointCount P i.2

Next tactic:
rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
    have step3 : ∑ i in sᶜ, lineCount L i.1 = ∑ i in sᶜ, pointCount P i.2 := by
      rwa [← s.sum_add_sum_compl, ← s.sum_add_sum_compl, step2, add_left_cancel_iff] at step1
    

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2
step3 : ∑ i in sᶜ, lineCount L i.1 = ∑ i in sᶜ, pointCount P i.2
⊢ lineCount L p = pointCount P l

Next tactic:
rw [← Set.toFinset_compl] at step3
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
    have step3 : ∑ i in sᶜ, lineCount L i.1 = ∑ i in 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
    

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2
step3 : ∑ i in Set.toFinset {i | i.1 ∈ i.2}ᶜ, lineCount L i.1 = ∑ i in Set.toFinset {i | i.1 ∈ i.2}ᶜ, pointCount P i.2
⊢ lineCount L p = pointCount P l

Next tactic:
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
theorem HasLines.lineCount_eq_pointCount [HasLines P L] [Fintype P] [Fintype L]
    (hPL : Fintype.card P = Fintype.card L) {p : P} {l : L} (hpl : p ∉ l) :
    lineCount L p = pointCount P l := by
  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 in s, lineCount L i.1 = ∑ i in s, pointCount P i.2 := by
      rw [s.sum_finset_product Finset.univ fun p => Set.toFinset { l | p ∈ l }]
      rw [s.sum_finset_product_right Finset.univ fun l => Set.toFinset { p | p ∈ l }]
      refine'
        (Finset.sum_bij (fun l _ => f l) (fun l _ => Finset.mem_univ (f l)) (fun l hl => _)
            (fun _ _ _ _ h => hf1.1 h) fun p _ => _).symm
      · simp_rw [Finset.sum_const, Set.toFinset_card, ← Nat.card_eq_fintype_card]
        change pointCount P l • pointCount P l = lineCount L (f l) • lineCount L (f l)
        rw [hf2]
      · obtain ⟨l, hl⟩ := hf1.2 p
        exact ⟨l, Finset.mem_univ l, hl.symm⟩
      all_goals simp_rw [Finset.mem_univ, true_and_iff, Set.mem_toFinset]; exact fun p => Iff.rfl
    have step3 : ∑ i in sᶜ, lineCount L i.1 = ∑ i in 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 

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
hPL : Fintype.card P = Fintype.card L
p : P
l : L
hpl : p ∉ l
f : L → P
hf1 : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
s : Finset (P × L) := Set.toFinset {i | i.1 ∈ i.2}
step1 : ∑ i : P × L, lineCount L i.1 = ∑ i : P × L, pointCount P i.2
step2 : ∑ i in s, lineCount L i.1 = ∑ i in s, pointCount P i.2
step3 : ∑ i in Set.toFinset {i | i.1 ∈ i.2}ᶜ, lineCount L i.1 = ∑ i in Set.toFinset {i | i.1 ∈ i.2}ᶜ, pointCount P i.2
i : P × L
hi : i ∈ Set.toFinset {i | i.1 ∈ i.2}ᶜ
⊢ i.1 ∉ i.2

Next tactic:
exact Set.mem_toFinset.mp hi
/-- If a nondegenerate configuration has a unique line through any two points, and if `|P| = |L|`,
  then there is a unique point on any two lines. -/
noncomputable def HasLines.hasPoints [HasLines P L] [Fintype P] [Fintype L]
    (h : Fintype.card P = Fintype.card L) : HasPoints P L :=
  let this : ∀ l₁ l₂ : L, l₁ ≠ l₂ → ∃ p : P, p ∈ l₁ ∧ p ∈ l₂ := fun l₁ l₂ hl => by
    

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
      have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
      obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
      by_cases hl₂ : p ∈ l₂
      exact ⟨p, hl₁, hl₂⟩
      have key' : Fintype.card { q : P // q ∈ l₂ } = Fintype.card { l : L // p ∈ l } :=
        ((HasLines.lineCount_eq_pointCount h hl₂).trans Nat.card_eq_fintype_card).symm.trans
          Nat.card_eq_fintype_card
      have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr q.2)
      let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
        ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
      have hf : Function.Injective f := fun q₁ q₂ hq =>
        Subtype.ext
          ((eq_or_eq q₁.2 q₂.2 (mkLine_ax (this q₁)).2
                ((congr_arg _ (Subtype.ext_iff.mp hq)).mpr (mkLine_ax (this q₂)).2)).resolve_right
            fun h => (congr_arg (¬p ∈ ·) h).mp hl₂ (mkLine_ax (this q₁)).1)
      have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).2
      obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
      exact ⟨q, (congr_arg _ (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
/-- If a nondegenerate configuration has a unique line through any two points, and if `|P| = |L|`,
  then there is a unique point on any two lines. -/
noncomputable def HasLines.hasPoints [HasLines P L] [Fintype P] [Fintype L]
    (h : Fintype.card P = Fintype.card L) : HasPoints P L :=
  let this : ∀ l₁ l₂ : L, l₁ ≠ l₂ → ∃ p : P, p ∈ l₁ ∧ p ∈ l₂ := fun l₁ l₂ hl => by
    classical
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
/-- If a nondegenerate configuration has a unique line through any two points, and if `|P| = |L|`,
  then there is a unique point on any two lines. -/
noncomputable def HasLines.hasPoints [HasLines P L] [Fintype P] [Fintype L]
    (h : Fintype.card P = Fintype.card L) : HasPoints P L :=
  let this : ∀ l₁ l₂ : L, l₁ ≠ l₂ → ∃ p : P, p ∈ l₁ ∧ p ∈ l₂ := fun l₁ l₂ hl => by
    classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left✝ : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
/-- If a nondegenerate configuration has a unique line through any two points, and if `|P| = |L|`,
  then there is a unique point on any two lines. -/
noncomputable def HasLines.hasPoints [HasLines P L] [Fintype P] [Fintype L]
    (h : Fintype.card P = Fintype.card L) : HasPoints P L :=
  let this : ∀ l₁ l₂ : L, l₁ ≠ l₂ → ∃ p : P, p ∈ l₁ ∧ p ∈ l₂ := fun l₁ l₂ hl => by
    classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left✝ : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this : Nontrivial L
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
/-- If a nondegenerate configuration has a unique line through any two points, and if `|P| = |L|`,
  then there is a unique point on any two lines. -/
noncomputable def HasLines.hasPoints [HasLines P L] [Fintype P] [Fintype L]
    (h : Fintype.card P = Fintype.card L) : HasPoints P L :=
  let this : ∀ l₁ l₂ : L, l₁ ≠ l₂ → ∃ p : P, p ∈ l₁ ∧ p ∈ l₂ := fun l₁ l₂ hl => by
    classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left✝ : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this✝ : Nontrivial L
this : Nontrivial P
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
/-- If a nondegenerate configuration has a unique line through any two points, and if `|P| = |L|`,
  then there is a unique point on any two lines. -/
noncomputable def HasLines.hasPoints [HasLines P L] [Fintype P] [Fintype L]
    (h : Fintype.card P = Fintype.card L) : HasPoints P L :=
  let this : ∀ l₁ l₂ : L, l₁ ≠ l₂ → ∃ p : P, p ∈ l₁ ∧ p ∈ l₂ := fun l₁ l₂ hl => by
    classical
      obtain ⟨f, _, hf2⟩ := HasLines.exists_bijective_of_card_eq h
      haveI : Nontrivial L := ⟨⟨l₁, l₂, hl⟩⟩
      haveI := Fintype.one_lt_card_iff_nontrivial.mp ((congr_arg _ h).mpr Fintype.one_lt_card)
      have h₁ : ∀ p : P, 0 < lineCount L p := fun p =>
        Exists.elim (exists_ne p) fun q hq =>
          (congr_arg _ Nat.card_eq_fintype_card).mpr
            (Fintype.card_pos_iff.mpr ⟨⟨mkLine hq, (mkLine_ax hq).2⟩⟩)
      

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : HasLines P L
inst✝¹ : Fintype P
inst✝ : Fintype L
h : Fintype.card P = Fintype.card L
l₁ l₂ : L
hl : l₁ ≠ l₂
f : L → P
left✝ : Function.Bijective f
hf2 : ∀ (l : L), pointCount P l = lineCount L (f l)
this✝ : Nontrivial L
this : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))