l3lab/ntp-mathlib huggingface dataset:
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
⊢ lineCount L p = lineCount L q

Next tactic:
have hp₂ : lineCount L p₂ = n := rfl
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
⊢ lineCount L p = lineCount L q

Next tactic:
have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
⊢ lineCount L p = lineCount L q

Next tactic:
have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
⊢ lineCount L p = lineCount L q

Next tactic:
have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
hl₃ : pointCount P l₃ = n
⊢ lineCount L p = lineCount L q

Next tactic:
have hp₁ : lineCount L p₁ = n := (HasLines.lineCount_eq_pointCount h h₁₃).trans hl₃
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
  have hp₁ : lineCount L p₁ = n := (HasLines.lineCount_eq_pointCount h h₁₃).trans hl₃
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
hl₃ : pointCount P l₃ = n
hp₁ : lineCount L p₁ = n
⊢ lineCount L p = lineCount L q

Next tactic:
have hl₂ : pointCount P l₂ = n := (HasLines.lineCount_eq_pointCount h h₁₂).symm.trans hp₁
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
  have hp₁ : lineCount L p₁ = n := (HasLines.lineCount_eq_pointCount h h₁₃).trans hl₃
  have hl₂ : pointCount P l₂ = n := (HasLines.lineCount_eq_pointCount h h₁₂).symm.trans hp₁
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
hl₃ : pointCount P l₃ = n
hp₁ : lineCount L p₁ = n
hl₂ : pointCount P l₂ = n
⊢ lineCount L p = lineCount L q

Next tactic:
suffices ∀ p : P, lineCount L p = n by exact (this p).trans (this q).symm
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
  have hp₁ : lineCount L p₁ = n := (HasLines.lineCount_eq_pointCount h h₁₃).trans hl₃
  have hl₂ : pointCount P l₂ = n := (HasLines.lineCount_eq_pointCount h h₁₂).symm.trans hp₁
  suffices ∀ p : P, lineCount L p = n by 

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
hl₃ : pointCount P l₃ = n
hp₁ : lineCount L p₁ = n
hl₂ : pointCount P l₂ = n
this : ∀ (p : P), lineCount L p = n
⊢ lineCount L p = lineCount L q

Next tactic:
exact (this p).trans (this q).symm
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
  have hp₁ : lineCount L p₁ = n := (HasLines.lineCount_eq_pointCount h h₁₃).trans hl₃
  have hl₂ : pointCount P l₂ = n := (HasLines.lineCount_eq_pointCount h h₁₂).symm.trans hp₁
  suffices ∀ p : P, lineCount L p = n by exact (this p).trans (this q).symm
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
hl₃ : pointCount P l₃ = n
hp₁ : lineCount L p₁ = n
hl₂ : pointCount P l₂ = n
⊢ ∀ (p : P), lineCount L p = n

Next tactic:
refine' fun p =>
    or_not.elim (fun h₂ => _) fun h₂ => (HasLines.lineCount_eq_pointCount h h₂).trans hl₂
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
  have hp₁ : lineCount L p₁ = n := (HasLines.lineCount_eq_pointCount h h₁₃).trans hl₃
  have hl₂ : pointCount P l₂ = n := (HasLines.lineCount_eq_pointCount h h₁₂).symm.trans hp₁
  suffices ∀ p : P, lineCount L p = n by exact (this p).trans (this q).symm
  refine' fun p =>
    or_not.elim (fun h₂ => _) fun h₂ => (HasLines.lineCount_eq_pointCount h h₂).trans hl₂
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p✝ q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
hl₃ : pointCount P l₃ = n
hp₁ : lineCount L p₁ = n
hl₂ : pointCount P l₂ = n
p : P
h₂ : p ∈ l₂
⊢ lineCount L p = n

Next tactic:
refine' or_not.elim (fun h₃ => _) fun h₃ => (HasLines.lineCount_eq_pointCount h h₃).trans hl₃
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := by
  cases nonempty_fintype P
  cases nonempty_fintype L
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  have h := card_points_eq_card_lines P L
  let n := lineCount L p₂
  have hp₂ : lineCount L p₂ = n := rfl
  have hl₁ : pointCount P l₁ = n := (HasLines.lineCount_eq_pointCount h h₂₁).symm.trans hp₂
  have hp₃ : lineCount L p₃ = n := (HasLines.lineCount_eq_pointCount h h₃₁).trans hl₁
  have hl₃ : pointCount P l₃ = n := (HasLines.lineCount_eq_pointCount h h₃₃).symm.trans hp₃
  have hp₁ : lineCount L p₁ = n := (HasLines.lineCount_eq_pointCount h h₁₃).trans hl₃
  have hl₂ : pointCount P l₂ = n := (HasLines.lineCount_eq_pointCount h h₁₂).symm.trans hp₁
  suffices ∀ p : P, lineCount L p = n by exact (this p).trans (this q).symm
  refine' fun p =>
    or_not.elim (fun h₂ => _) fun h₂ => (HasLines.lineCount_eq_pointCount h h₂).trans hl₂
  refine' or_not.elim (fun h₃ => _) fun h₃ => (HasLines.lineCount_eq_pointCount h h₃).trans hl₃
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p✝ q : P
val✝¹ : Fintype P
val✝ : Fintype L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₁₂ : p₁ ∉ l₂
h₁₃ : p₁ ∉ l₃
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
h : Fintype.card P = Fintype.card L
n : ℕ := lineCount L p₂
hp₂ : lineCount L p₂ = n
hl₁ : pointCount P l₁ = n
hp₃ : lineCount L p₃ = n
hl₃ : pointCount P l₃ = n
hp₁ : lineCount L p₁ = n
hl₂ : pointCount P l₂ = n
p : P
h₂ : p ∈ l₂
h₃ : p ∈ l₃
⊢ lineCount L p = n

Next tactic:
rw [(eq_or_eq h₂ h₂₂ h₃ h₂₃).resolve_right fun h =>
      h₃₃ ((congr_arg (Membership.mem p₃) h).mp h₃₂)]
theorem pointCount_eq_pointCount [Finite P] [Finite L] (l m : L) :
    pointCount P l = pointCount P m := by
  

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
l m : L
⊢ pointCount P l = pointCount P m

Next tactic:
apply lineCount_eq_lineCount (Dual P)
theorem lineCount_eq_pointCount [Finite P] [Finite L] (p : P) (l : L) :
    lineCount L p = pointCount P l :=
  Exists.elim (exists_point l) fun q hq =>
    (lineCount_eq_lineCount L p q).trans <| by
      

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p : P
l : L
q : P
hq : q ∉ l
⊢ lineCount L q = pointCount P l

Next tactic:
cases nonempty_fintype P
theorem lineCount_eq_pointCount [Finite P] [Finite L] (p : P) (l : L) :
    lineCount L p = pointCount P l :=
  Exists.elim (exists_point l) fun q hq =>
    (lineCount_eq_lineCount L p q).trans <| by
      cases nonempty_fintype P
      

Proof state:
case intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p : P
l : L
q : P
hq : q ∉ l
val✝ : Fintype P
⊢ lineCount L q = pointCount P l

Next tactic:
cases nonempty_fintype L
theorem lineCount_eq_pointCount [Finite P] [Finite L] (p : P) (l : L) :
    lineCount L p = pointCount P l :=
  Exists.elim (exists_point l) fun q hq =>
    (lineCount_eq_lineCount L p q).trans <| by
      cases nonempty_fintype P
      cases nonempty_fintype L
      

Proof state:
case intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p : P
l : L
q : P
hq : q ∉ l
val✝¹ : Fintype P
val✝ : Fintype L
⊢ lineCount L q = pointCount P l

Next tactic:
exact HasLines.lineCount_eq_pointCount (card_points_eq_card_lines P L) hq
theorem lineCount_eq [Finite P] [Finite L] (p : P) : lineCount L p = order P L + 1 := by
  

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p : P
⊢ lineCount L p = order P L + 1

Next tactic:
classical
    obtain ⟨q, -, -, l, -, -, -, -, h, -⟩ := Classical.choose_spec (@exists_config P L _ _)
    cases nonempty_fintype { l : L // q ∈ l }
    rw [order, lineCount_eq_lineCount L p q, lineCount_eq_lineCount L (Classical.choose _) q,
      lineCount, Nat.card_eq_fintype_card, Nat.sub_add_cancel]
    exact Fintype.card_pos_iff.mpr ⟨⟨l, h⟩⟩
theorem lineCount_eq [Finite P] [Finite L] (p : P) : lineCount L p = order P L + 1 := by
  classical
    

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p : P
⊢ lineCount L p = order P L + 1

Next tactic:
obtain ⟨q, -, -, l, -, -, -, -, h, -⟩ := Classical.choose_spec (@exists_config P L _ _)
theorem lineCount_eq [Finite P] [Finite L] (p : P) : lineCount L p = order P L + 1 := by
  classical
    obtain ⟨q, -, -, l, -, -, -, -, h, -⟩ := Classical.choose_spec (@exists_config P L _ _)
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
l : L
h : q ∈ l
⊢ lineCount L p = order P L + 1

Next tactic:
cases nonempty_fintype { l : L // q ∈ l }
theorem lineCount_eq [Finite P] [Finite L] (p : P) : lineCount L p = order P L + 1 := by
  classical
    obtain ⟨q, -, -, l, -, -, -, -, h, -⟩ := Classical.choose_spec (@exists_config P L _ _)
    cases nonempty_fintype { l : L // q ∈ l }
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
l : L
h : q ∈ l
val✝ : Fintype { l // q ∈ l }
⊢ lineCount L p = order P L + 1

Next tactic:
rw [order, lineCount_eq_lineCount L p q, lineCount_eq_lineCount L (Classical.choose _) q,
      lineCount, Nat.card_eq_fintype_card, Nat.sub_add_cancel]
theorem lineCount_eq [Finite P] [Finite L] (p : P) : lineCount L p = order P L + 1 := by
  classical
    obtain ⟨q, -, -, l, -, -, -, -, h, -⟩ := Classical.choose_spec (@exists_config P L _ _)
    cases nonempty_fintype { l : L // q ∈ l }
    rw [order, lineCount_eq_lineCount L p q, lineCount_eq_lineCount L (Classical.choose _) q,
      lineCount, Nat.card_eq_fintype_card, Nat.sub_add_cancel]
    

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p q : P
l : L
h : q ∈ l
val✝ : Fintype { l // q ∈ l }
⊢ 1 ≤ Fintype.card { l // q ∈ l }

Next tactic:
exact Fintype.card_pos_iff.mpr ⟨⟨l, h⟩⟩
theorem one_lt_order [Finite P] [Finite L] : 1 < order P L := by
  

Proof state:
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
⊢ 1 < order P L

Next tactic:
obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, -, -, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
theorem one_lt_order [Finite P] [Finite L] : 1 < order P L := by
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, -, -, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
⊢ 1 < order P L

Next tactic:
cases nonempty_fintype { p : P // p ∈ l₂ }
theorem one_lt_order [Finite P] [Finite L] : 1 < order P L := by
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, -, -, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  cases nonempty_fintype { p : P // p ∈ l₂ }
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
val✝ : Fintype { p // p ∈ l₂ }
⊢ 1 < order P L

Next tactic:
rw [← add_lt_add_iff_right 1, ← pointCount_eq _ l₂, pointCount, Nat.card_eq_fintype_card,
    Fintype.two_lt_card_iff]
theorem one_lt_order [Finite P] [Finite L] : 1 < order P L := by
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, -, -, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  cases nonempty_fintype { p : P // p ∈ l₂ }
  rw [← add_lt_add_iff_right 1, ← pointCount_eq _ l₂, pointCount, Nat.card_eq_fintype_card,
    Fintype.two_lt_card_iff]
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
val✝ : Fintype { p // p ∈ l₂ }
⊢ ∃ a b c, a ≠ b ∧ a ≠ c ∧ b ≠ c

Next tactic:
simp_rw [Ne, Subtype.ext_iff]
theorem one_lt_order [Finite P] [Finite L] : 1 < order P L := by
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, -, -, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  cases nonempty_fintype { p : P // p ∈ l₂ }
  rw [← add_lt_add_iff_right 1, ← pointCount_eq _ l₂, pointCount, Nat.card_eq_fintype_card,
    Fintype.two_lt_card_iff]
  simp_rw [Ne, Subtype.ext_iff]
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
val✝ : Fintype { p // p ∈ l₂ }
⊢ ∃ a b c, ¬↑a = ↑b ∧ ¬↑a = ↑c ∧ ¬↑b = ↑c

Next tactic:
have h := mkPoint_ax fun h => h₂₁ ((congr_arg _ h).mpr h₂₂)
theorem one_lt_order [Finite P] [Finite L] : 1 < order P L := by
  obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, -, -, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
  cases nonempty_fintype { p : P // p ∈ l₂ }
  rw [← add_lt_add_iff_right 1, ← pointCount_eq _ l₂, pointCount, Nat.card_eq_fintype_card,
    Fintype.two_lt_card_iff]
  simp_rw [Ne, Subtype.ext_iff]
  have h := mkPoint_ax fun h => h₂₁ ((congr_arg _ h).mpr h₂₂)
  

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro
P : Type u_1
L : Type u_2
inst✝³ : Membership P L
inst✝² : ProjectivePlane P L
inst✝¹ : Finite P
inst✝ : Finite L
p₁ p₂ p₃ : P
l₁ l₂ l₃ : L
h₂₁ : p₂ ∉ l₁
h₂₂ : p₂ ∈ l₂
h₂₃ : p₂ ∈ l₃
h₃₁ : p₃ ∉ l₁
h₃₂ : p₃ ∈ l₂
h₃₃ : p₃ ∉ l₃
val✝ : Fintype { p // p ∈ l₂ }
h : mkPoint (_ : l₁ = l₂ → False) ∈ l₁ ∧ mkPoint (_ : l₁ = l₂ → False) ∈ l₂
⊢ ∃ a b c, ¬↑a = ↑b ∧ ¬↑a = ↑c ∧ ¬↑b = ↑c

Next tactic:
exact
    ⟨⟨mkPoint _, h.2⟩, ⟨p₂, h₂₂⟩, ⟨p₃, h₃₂⟩, ne_of_mem_of_not_mem h.1 h₂₁,
      ne_of_mem_of_not_mem h.1 h₃₁, ne_of_mem_of_not_mem h₂₃ h₃₃⟩