l3lab/ntp-mathlib huggingface dataset:
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by 

Proof state:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hv : v ∈ direction s
⊢ Finset.univ = {0, 1}

Next tactic:
decide
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide
      

Proof state:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hv : v ∈ direction s
hu : Finset.univ = {0, 1}
⊢ Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁}

Next tactic:
rw [← Fintype.coe_image_univ, hu]
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide
      rw [← Fintype.coe_image_univ, hu]
      

Proof state:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hv : v ∈ direction s
hu : Finset.univ = {0, 1}
⊢ ↑(Finset.image b {0, 1}) = {c₂ -ᵥ c₁, p₂ -ᵥ p₁}

Next tactic:
simp
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide
      rw [← Fintype.coe_image_univ, hu]
      simp
    

Proof state:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hv : v ∈ direction s
hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁}
⊢ ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)

Next tactic:
rw [← hbs, hr, Submodule.mem_span_insert] at hv
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide
      rw [← Fintype.coe_image_univ, hu]
      simp
    rw [← hbs, hr, Submodule.mem_span_insert] at hv
    

Proof state:
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hv : ∃ a, ∃ z ∈ Submodule.span ℝ {p₂ -ᵥ p₁}, v = a • (c₂ -ᵥ c₁) + z
hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁}
⊢ ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)

Next tactic:
rcases hv with ⟨t₁, v', hv', hv⟩
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide
      rw [← Fintype.coe_image_univ, hu]
      simp
    rw [← hbs, hr, Submodule.mem_span_insert] at hv
    rcases hv with ⟨t₁, v', hv', hv⟩
    

Proof state:
case intro.intro.intro
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁}
t₁ : ℝ
v' : V
hv' : v' ∈ Submodule.span ℝ {p₂ -ᵥ p₁}
hv : v = t₁ • (c₂ -ᵥ c₁) + v'
⊢ ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)

Next tactic:
rw [Submodule.mem_span_singleton] at hv'
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide
      rw [← Fintype.coe_image_univ, hu]
      simp
    rw [← hbs, hr, Submodule.mem_span_insert] at hv
    rcases hv with ⟨t₁, v', hv', hv⟩
    rw [Submodule.mem_span_singleton] at hv'
    

Proof state:
case intro.intro.intro
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁}
t₁ : ℝ
v' : V
hv' : ∃ a, a • (p₂ -ᵥ p₁) = v'
hv : v = t₁ • (c₂ -ᵥ c₁) + v'
⊢ ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)

Next tactic:
rcases hv' with ⟨t₂, rfl⟩
/-- Distances `r₁` `r₂` of `p` from two different points `c₁` `c₂` determine at
most two points `p₁` `p₂` in a two-dimensional subspace containing those points
(two circles intersect in at most two points). -/
theorem eq_of_dist_eq_of_dist_eq_of_mem_of_finrank_eq_two {s : AffineSubspace ℝ P}
    [FiniteDimensional ℝ s.direction] (hd : finrank ℝ s.direction = 2) {c₁ c₂ p₁ p₂ p : P}
    (hc₁s : c₁ ∈ s) (hc₂s : c₂ ∈ s) (hp₁s : p₁ ∈ s) (hp₂s : p₂ ∈ s) (hps : p ∈ s) {r₁ r₂ : ℝ}
    (hc : c₁ ≠ c₂) (hp : p₁ ≠ p₂) (hp₁c₁ : dist p₁ c₁ = r₁) (hp₂c₁ : dist p₂ c₁ = r₁)
    (hpc₁ : dist p c₁ = r₁) (hp₁c₂ : dist p₁ c₂ = r₂) (hp₂c₂ : dist p₂ c₂ = r₂)
    (hpc₂ : dist p c₂ = r₂) : p = p₁ ∨ p = p₂ := by
  have ho : ⟪c₂ -ᵥ c₁, p₂ -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hp₂c₁.symm) (hp₁c₂.trans hp₂c₂.symm)
  have hop : ⟪c₂ -ᵥ c₁, p -ᵥ p₁⟫ = 0 :=
    inner_vsub_vsub_of_dist_eq_of_dist_eq (hp₁c₁.trans hpc₁.symm) (hp₁c₂.trans hpc₂.symm)
  let b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
  have hb : LinearIndependent ℝ b := by
    refine' linearIndependent_of_ne_zero_of_inner_eq_zero _ _
    · intro i
      fin_cases i <;> simp [hc.symm, hp.symm]
    · intro i j hij
      fin_cases i <;> fin_cases j <;> try exact False.elim (hij rfl)
      · exact ho
      · rw [real_inner_comm]
        exact ho
  have hbs : Submodule.span ℝ (Set.range b) = s.direction := by
    refine' eq_of_le_of_finrank_eq _ _
    · rw [Submodule.span_le, Set.range_subset_iff]
      intro i
      fin_cases i
      · exact vsub_mem_direction hc₂s hc₁s
      · exact vsub_mem_direction hp₂s hp₁s
    · rw [finrank_span_eq_card hb, Fintype.card_fin, hd]
  have hv : ∀ v ∈ s.direction, ∃ t₁ t₂ : ℝ, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁) := by
    intro v hv
    have hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁} := by
      have hu : (Finset.univ : Finset (Fin 2)) = {0, 1} := by decide
      rw [← Fintype.coe_image_univ, hu]
      simp
    rw [← hbs, hr, Submodule.mem_span_insert] at hv
    rcases hv with ⟨t₁, v', hv', hv⟩
    rw [Submodule.mem_span_singleton] at hv'
    rcases hv' with ⟨t₂, rfl⟩
    

Proof state:
case intro.intro.intro.intro
V : Type u_1
P : Type u_2
inst✝⁴ : NormedAddCommGroup V
inst✝³ : InnerProductSpace ℝ V
inst✝² : MetricSpace P
inst✝¹ : NormedAddTorsor V P
s : AffineSubspace ℝ P
inst✝ : FiniteDimensional ℝ ↥(direction s)
hd : finrank ℝ ↥(direction s) = 2
c₁ c₂ p₁ p₂ p : P
hc₁s : c₁ ∈ s
hc₂s : c₂ ∈ s
hp₁s : p₁ ∈ s
hp₂s : p₂ ∈ s
hps : p ∈ s
r₁ r₂ : ℝ
hc : c₁ ≠ c₂
hp : p₁ ≠ p₂
hp₁c₁ : dist p₁ c₁ = r₁
hp₂c₁ : dist p₂ c₁ = r₁
hpc₁ : dist p c₁ = r₁
hp₁c₂ : dist p₁ c₂ = r₂
hp₂c₂ : dist p₂ c₂ = r₂
hpc₂ : dist p c₂ = r₂
ho : inner (c₂ -ᵥ c₁) (p₂ -ᵥ p₁) = 0
hop : inner (c₂ -ᵥ c₁) (p -ᵥ p₁) = 0
b : Fin 2 → V := ![c₂ -ᵥ c₁, p₂ -ᵥ p₁]
hb : LinearIndependent ℝ b
hbs : Submodule.span ℝ (Set.range b) = direction s
v : V
hr : Set.range b = {c₂ -ᵥ c₁, p₂ -ᵥ p₁}
t₁ t₂ : ℝ
hv : v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)
⊢ ∃ t₁ t₂, v = t₁ • (c₂ -ᵥ c₁) + t₂ • (p₂ -ᵥ p₁)

Next tactic:
exact ⟨t₁, t₂, hv⟩