l3lab/ntp-mathlib huggingface dataset:
/-- 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⟩⟩)
      have h₂ : ∀ l : L, 0 < pointCount P l := fun l => (congr_arg _ (hf2 l)).mpr (h₁ (f l))
      

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
h₂ : ∀ (l : L), 0 < pointCount P l
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
obtain ⟨p, hl₁⟩ := Fintype.card_pos_iff.mp ((congr_arg _ Nat.card_eq_fintype_card).mp (h₂ l₁))
/-- 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⟩⟩)
      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))
      

Proof state:
case intro.intro.intro.mk
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
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
by_cases hl₂ : p ∈ l₂
/-- 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⟩⟩)
      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
      

Proof state:
case pos
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
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∈ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂
case neg
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
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
exact ⟨p, hl₁, hl₂⟩
/-- 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⟩⟩)
      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⟩
      

Proof state:
case neg
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
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
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
/-- 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⟩⟩)
      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
      

Proof state:
case neg
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
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key' : Fintype.card { q // q ∈ l₂ } = Fintype.card { l // p ∈ l }
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
have : ∀ q : { q // q ∈ l₂ }, p ≠ q := fun q hq => hl₂ ((congr_arg (· ∈ l₂) hq).mpr 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
      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)
      

Proof state:
case neg
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✝¹ : Nontrivial L
this✝ : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key' : Fintype.card { q // q ∈ l₂ } = Fintype.card { l // p ∈ l }
this : ∀ (q : { q // q ∈ l₂ }), p ≠ ↑q
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
let f : { q : P // q ∈ l₂ } → { l : L // p ∈ l } := fun q =>
        ⟨mkLine (this q), (mkLine_ax (this q)).1⟩
/-- 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⟩⟩)
      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⟩
      

Proof state:
case neg
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✝¹ : Nontrivial L
this✝ : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key' : Fintype.card { q // q ∈ l₂ } = Fintype.card { l // p ∈ l }
this : ∀ (q : { q // q ∈ l₂ }), p ≠ ↑q
f : { q // q ∈ l₂ } → { l // p ∈ l } :=
  fun q => { val := mkLine (_ : p ≠ ↑q), property := (_ : p ∈ mkLine (_ : p ≠ ↑q)) }
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
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)
/-- 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⟩⟩)
      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)
      

Proof state:
case neg
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✝¹ : Nontrivial L
this✝ : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key' : Fintype.card { q // q ∈ l₂ } = Fintype.card { l // p ∈ l }
this : ∀ (q : { q // q ∈ l₂ }), p ≠ ↑q
f : { q // q ∈ l₂ } → { l // p ∈ l } :=
  fun q => { val := mkLine (_ : p ≠ ↑q), property := (_ : p ∈ mkLine (_ : p ≠ ↑q)) }
hf : Function.Injective f
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
have key' := ((Fintype.bijective_iff_injective_and_card f).mpr ⟨hf, key'⟩).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⟩⟩)
      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
      

Proof state:
case neg
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✝¹ : Nontrivial L
this✝ : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key'✝ : Fintype.card { q // q ∈ l₂ } = Fintype.card { l // p ∈ l }
this : ∀ (q : { q // q ∈ l₂ }), p ≠ ↑q
f : { q // q ∈ l₂ } → { l // p ∈ l } :=
  fun q => { val := mkLine (_ : p ≠ ↑q), property := (_ : p ∈ mkLine (_ : p ≠ ↑q)) }
hf : Function.Injective f
key' : Function.Surjective f
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
obtain ⟨q, hq⟩ := key' ⟨l₁, hl₁⟩
/-- 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⟩⟩)
      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⟩
      

Proof state:
case neg.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✝¹ : Nontrivial L
this✝ : Nontrivial P
h₁ : ∀ (p : P), 0 < lineCount L p
h₂ : ∀ (l : L), 0 < pointCount P l
p : P
hl₁ : p ∈ l₁
hl₂ : p ∉ l₂
key'✝ : Fintype.card { q // q ∈ l₂ } = Fintype.card { l // p ∈ l }
this : ∀ (q : { q // q ∈ l₂ }), p ≠ ↑q
f : { q // q ∈ l₂ } → { l // p ∈ l } :=
  fun q => { val := mkLine (_ : p ≠ ↑q), property := (_ : p ∈ mkLine (_ : p ≠ ↑q)) }
hf : Function.Injective f
key' : Function.Surjective f
q : { q // q ∈ l₂ }
hq : f q = { val := l₁, property := hl₁ }
⊢ ∃ p ∈ l₁, p ∈ l₂

Next tactic:
exact ⟨q, (congr_arg _ (Subtype.ext_iff.mp hq)).mp (mkLine_ax (this q)).2, q.2⟩
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := 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
⊢ lineCount L p = lineCount L q

Next tactic:
cases nonempty_fintype P
theorem lineCount_eq_lineCount [Finite P] [Finite L] (p q : P) : lineCount L p = lineCount L q := 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 q : P
val✝ : Fintype P
⊢ lineCount L p = lineCount L q

Next tactic:
cases nonempty_fintype L
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
  

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 q : P
val✝¹ : Fintype P
val✝ : Fintype L
⊢ lineCount L p = lineCount L q

Next tactic:
obtain ⟨p₁, p₂, p₃, l₁, l₂, l₃, h₁₂, h₁₃, h₂₁, h₂₂, h₂₃, h₃₁, h₃₂, h₃₃⟩ := @exists_config P L _ _
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 _ _
  

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
⊢ lineCount L p = lineCount L q

Next tactic:
have h := card_points_eq_card_lines P L
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
  

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
⊢ lineCount L p = lineCount L q

Next tactic:
let n := lineCount L p₂