l3lab/ntp-mathlib huggingface dataset:
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖)) := by
      ring
    have H2 :
      ⟪x, x⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, x⟫ - ⟪x, y⟫) * (⟪x, x⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    have H3 :
      ⟪y, y⟫ * (⟪y, y⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪x, x⟫)) - (⟪y, y⟫ - ⟪x, y⟫) * (⟪y, y⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖))
H2 :
  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =
    inner x x * inner y y - inner x y * inner x y
H3 :
  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =
    inner x x * inner y y - inner x y * inner x y
⊢ (inner x (x - y) / (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) -
              Real.sin (angle x (x - y)) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    -(inner x y / (‖x‖ * ‖y‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
rw [mul_sub_right_distrib, mul_sub_right_distrib, mul_sub_right_distrib, mul_sub_right_distrib,
      H1, sin_angle_mul_norm_mul_norm, norm_sub_rev x y, sin_angle_mul_norm_mul_norm,
      norm_sub_rev y x, inner_sub_left, inner_sub_left, inner_sub_right, inner_sub_right,
      inner_sub_right, inner_sub_right, real_inner_comm x y, H2, H3,
      Real.mul_self_sqrt (sub_nonneg_of_le (real_inner_mul_inner_self_le x y)),
      real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm,
      real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]
/-- The cosine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) *
          (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖)) := by
      ring
    have H2 :
      ⟪x, x⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, x⟫ - ⟪x, y⟫) * (⟪x, x⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    have H3 :
      ⟪y, y⟫ * (⟪y, y⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪x, x⟫)) - (⟪y, y⟫ - ⟪x, y⟫) * (⟪y, y⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring
    rw [mul_sub_right_distrib, mul_sub_right_distrib, mul_sub_right_distrib, mul_sub_right_distrib,
      H1, sin_angle_mul_norm_mul_norm, norm_sub_rev x y, sin_angle_mul_norm_mul_norm,
      norm_sub_rev y x, inner_sub_left, inner_sub_left, inner_sub_right, inner_sub_right,
      inner_sub_right, inner_sub_right, real_inner_comm x y, H2, H3,
      Real.mul_self_sqrt (sub_nonneg_of_le (real_inner_mul_inner_self_le x y)),
      real_inner_self_eq_norm_mul_norm, real_inner_self_eq_norm_mul_norm,
      real_inner_eq_norm_mul_self_add_norm_mul_self_sub_norm_sub_mul_self_div_two]
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖x - y‖))
H2 :
  inner x x * (inner x x - inner x y - (inner x y - inner y y)) - (inner x x - inner x y) * (inner x x - inner x y) =
    inner x x * inner y y - inner x y * inner x y
H3 :
  inner y y * (inner y y - inner x y - (inner x y - inner x x)) - (inner y y - inner x y) * (inner y y - inner x y) =
    inner x x * inner y y - inner x y * inner x y
⊢ (‖x‖ * ‖x‖ - (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2) / (‖x‖ * ‖x - y‖) *
                ((‖y‖ * ‖y‖ - (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2) / (‖y‖ * ‖x - y‖)) *
              ‖x‖ *
            ‖y‖ *
          ‖x - y‖ *
        ‖x - y‖ -
      (‖x‖ * ‖x‖ * (‖y‖ * ‖y‖) -
        (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 * ((‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2)) =
    -((‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2 / (‖x‖ * ‖y‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
field_simp [hxn, hyn, hxyn]
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
⊢ Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)

Next tactic:
by_cases hxy : x = y
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · 

Proof state:
case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : x = y
⊢ Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)

Next tactic:
rw [hxy, angle_self hy]
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    

Proof state:
case pos
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : x = y
⊢ Real.sin (angle y (y - y) + angle y (y - y)) = Real.sin 0

Next tactic:
simp
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · 

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
⊢ Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y)

Next tactic:
rw [Real.sin_add, cos_angle, cos_angle]
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
      inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
    Real.sin (angle x y)

Next tactic:
have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
      inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
    Real.sin (angle x y)

Next tactic:
have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
      inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
    Real.sin (angle x y)

Next tactic:
have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
      inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x)) =
    Real.sin (angle x y)

Next tactic:
apply mul_right_cancel₀ hxn
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
        inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
      ‖x‖ =
    Real.sin (angle x y) * ‖x‖

Next tactic:
apply mul_right_cancel₀ hyn
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
          inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
        ‖x‖ *
      ‖y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖

Next tactic:
apply mul_right_cancel₀ hxyn
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
            inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
          ‖x‖ *
        ‖y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖

Next tactic:
apply mul_right_cancel₀ hxyn
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
⊢ Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖

Next tactic:
ring
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
have H2 :
      ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
        ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
      ring
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
    have H2 :
      ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
        ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
      

Proof state:
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
⊢ inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖

Next tactic:
ring
/-- The sine of the sum of two angles in a possibly degenerate
triangle (where two given sides are nonzero), vector angle form. -/
theorem sin_angle_sub_add_angle_sub_rev_eq_sin_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.sin (angle x (x - y) + angle y (y - x)) = Real.sin (angle x y) := by
  by_cases hxy : x = y
  · rw [hxy, angle_self hy]
    simp
  · rw [Real.sin_add, cos_angle, cos_angle]
    have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
    have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
    have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
    apply mul_right_cancel₀ hxn
    apply mul_right_cancel₀ hyn
    apply mul_right_cancel₀ hxyn
    apply mul_right_cancel₀ hxyn
    have H1 :
      Real.sin (angle x (x - y)) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
        Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (⟪y, y - x⟫ / (‖y‖ * ‖y - x‖)) * ‖y‖ := by
      ring
    have H2 :
      ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
        ⟪x, x - y⟫ / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖ := by
      ring
    

Proof state:
case neg
V : Type u_1
inst✝¹ : NormedAddCommGroup V
inst✝ : InnerProductSpace ℝ V
x y : V
hx : x ≠ 0
hy : y ≠ 0
hxy : ¬x = y
hxn : ‖x‖ ≠ 0
hyn : ‖y‖ ≠ 0
hxyn : ‖x - y‖ ≠ 0
H1 :
  Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖x‖ * ‖y‖ * ‖x - y‖ =
    Real.sin (angle x (x - y)) * (‖x‖ * ‖x - y‖) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) * ‖y‖
H2 :
  inner x (x - y) / (‖x‖ * ‖y - x‖) * Real.sin (angle y (y - x)) * ‖x‖ * ‖y‖ * ‖y - x‖ =
    inner x (x - y) / (‖x‖ * ‖y - x‖) * (Real.sin (angle y (y - x)) * (‖y‖ * ‖y - x‖)) * ‖x‖
⊢ (Real.sin (angle x (x - y)) * (inner y (y - x) / (‖y‖ * ‖y - x‖)) +
              inner x (x - y) / (‖x‖ * ‖x - y‖) * Real.sin (angle y (y - x))) *
            ‖x‖ *
          ‖y‖ *
        ‖x - y‖ *
      ‖x - y‖ =
    Real.sin (angle x y) * ‖x‖ * ‖y‖ * ‖x - y‖ * ‖x - y‖

Next tactic:
have H3 :
      ⟪x, x⟫ * (⟪x, x⟫ - ⟪x, y⟫ - (⟪x, y⟫ - ⟪y, y⟫)) - (⟪x, x⟫ - ⟪x, y⟫) * (⟪x, x⟫ - ⟪x, y⟫) =
        ⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫ := by
      ring