File: Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean

full_name: Orientation._aux_Mathlib_Geometry_Euclidean_Angle_Oriented_Basic___macroRules_Orientation_termω_1
code: local notation "ω" => o.areaForm
goal: Lean.Macro

full_name: Orientation.oangle
code: /-- The oriented angle from `x` to `y`, modulo `2 * π`. If either vector is 0, this is 0.
See `InnerProductGeometry.angle` for the corresponding unoriented angle definition. -/
def oangle (x y : V) : Real.Angle 
goal: Real.Angle

full_name: Orientation.continuousAt_oangle
code: /-- Oriented angles are continuous when the vectors involved are nonzero. -/
theorem continuousAt_oangle {x : V × V} (hx1 : x.1 ≠ 0) (hx2 : x.2 ≠ 0) :
    ContinuousAt (fun y : V × V => o.oangle y.1 y.2) x 
goal: ContinuousAt (fun y => o.oangle y.1 y.2) x

full_name: Orientation.oangle_zero_left
code: /-- If the first vector passed to `oangle` is 0, the result is 0. -/
@[simp]
theorem oangle_zero_left (x : V) : o.oangle 0 x = 0 
goal: o.oangle 0 x = 0

full_name: Orientation.oangle_zero_right
code: /-- If the second vector passed to `oangle` is 0, the result is 0. -/
@[simp]
theorem oangle_zero_right (x : V) : o.oangle x 0 = 0 
goal: o.oangle x 0 = 0

full_name: Orientation.oangle_self
code: /-- If the two vectors passed to `oangle` are the same, the result is 0. -/
@[simp]
theorem oangle_self (x : V) : o.oangle x x = 0 
goal: o.oangle x x = 0

full_name: Orientation.left_ne_zero_of_oangle_ne_zero
code: /-- If the angle between two vectors is nonzero, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ 0 
goal: x ≠ 0

full_name: Orientation.right_ne_zero_of_oangle_ne_zero
code: /-- If the angle between two vectors is nonzero, the second vector is nonzero. -/
theorem right_ne_zero_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : y ≠ 0 
goal: y ≠ 0

full_name: Orientation.ne_of_oangle_ne_zero
code: /-- If the angle between two vectors is nonzero, the vectors are not equal. -/
theorem ne_of_oangle_ne_zero {x y : V} (h : o.oangle x y ≠ 0) : x ≠ y 
goal: x ≠ y

full_name: Orientation.left_ne_zero_of_oangle_eq_pi
code: /-- If the angle between two vectors is `π`, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_eq_pi {x y : V} (h : o.oangle x y = π) : x ≠ 0 
goal: x ≠ 0

full_name: Orientation.right_ne_zero_of_oangle_eq_pi
code: /-- If the angle between two vectors is `π`, the second vector is nonzero. -/
theorem right_ne_zero_of_oangle_eq_pi {x y : V} (h : o.oangle x y = π) : y ≠ 0 
goal: y ≠ 0

full_name: Orientation.ne_of_oangle_eq_pi
code: /-- If the angle between two vectors is `π`, the vectors are not equal. -/
theorem ne_of_oangle_eq_pi {x y : V} (h : o.oangle x y = π) : x ≠ y 
goal: x ≠ y

full_name: Orientation.left_ne_zero_of_oangle_eq_pi_div_two
code: /-- If the angle between two vectors is `π / 2`, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) : x ≠ 0 
goal: x ≠ 0

full_name: Orientation.right_ne_zero_of_oangle_eq_pi_div_two
code: /-- If the angle between two vectors is `π / 2`, the second vector is nonzero. -/
theorem right_ne_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) : y ≠ 0 
goal: y ≠ 0

full_name: Orientation.ne_of_oangle_eq_pi_div_two
code: /-- If the angle between two vectors is `π / 2`, the vectors are not equal. -/
theorem ne_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) : x ≠ y 
goal: x ≠ y

full_name: Orientation.left_ne_zero_of_oangle_eq_neg_pi_div_two
code: /-- If the angle between two vectors is `-π / 2`, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    x ≠ 0 
goal: x ≠ 0

full_name: Orientation.right_ne_zero_of_oangle_eq_neg_pi_div_two
code: /-- If the angle between two vectors is `-π / 2`, the second vector is nonzero. -/
theorem right_ne_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    y ≠ 0 
goal: y ≠ 0

full_name: Orientation.ne_of_oangle_eq_neg_pi_div_two
code: /-- If the angle between two vectors is `-π / 2`, the vectors are not equal. -/
theorem ne_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) : x ≠ y 
goal: x ≠ y

full_name: Orientation.left_ne_zero_of_oangle_sign_ne_zero
code: /-- If the sign of the angle between two vectors is nonzero, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_sign_ne_zero {x y : V} (h : (o.oangle x y).sign ≠ 0) : x ≠ 0 
goal: x ≠ 0

full_name: Orientation.right_ne_zero_of_oangle_sign_ne_zero
code: /-- If the sign of the angle between two vectors is nonzero, the second vector is nonzero. -/
theorem right_ne_zero_of_oangle_sign_ne_zero {x y : V} (h : (o.oangle x y).sign ≠ 0) : y ≠ 0 
goal: y ≠ 0

full_name: Orientation.ne_of_oangle_sign_ne_zero
code: /-- If the sign of the angle between two vectors is nonzero, the vectors are not equal. -/
theorem ne_of_oangle_sign_ne_zero {x y : V} (h : (o.oangle x y).sign ≠ 0) : x ≠ y 
goal: x ≠ y

full_name: Orientation.left_ne_zero_of_oangle_sign_eq_one
code: /-- If the sign of the angle between two vectors is positive, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) : x ≠ 0 
goal: x ≠ 0

full_name: Orientation.right_ne_zero_of_oangle_sign_eq_one
code: /-- If the sign of the angle between two vectors is positive, the second vector is nonzero. -/
theorem right_ne_zero_of_oangle_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) : y ≠ 0 
goal: y ≠ 0

full_name: Orientation.ne_of_oangle_sign_eq_one
code: /-- If the sign of the angle between two vectors is positive, the vectors are not equal. -/
theorem ne_of_oangle_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) : x ≠ y 
goal: x ≠ y

full_name: Orientation.left_ne_zero_of_oangle_sign_eq_neg_one
code: /-- If the sign of the angle between two vectors is negative, the first vector is nonzero. -/
theorem left_ne_zero_of_oangle_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) : x ≠ 0 
goal: x ≠ 0

full_name: Orientation.right_ne_zero_of_oangle_sign_eq_neg_one
code: /-- If the sign of the angle between two vectors is negative, the second vector is nonzero. -/
theorem right_ne_zero_of_oangle_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) : y ≠ 0 
goal: y ≠ 0

full_name: Orientation.ne_of_oangle_sign_eq_neg_one
code: /-- If the sign of the angle between two vectors is negative, the vectors are not equal. -/
theorem ne_of_oangle_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) : x ≠ y 
goal: x ≠ y

full_name: Orientation.oangle_rev
code: /-- Swapping the two vectors passed to `oangle` negates the angle. -/
theorem oangle_rev (x y : V) : o.oangle y x = -o.oangle x y 
goal: o.oangle y x = -o.oangle x y

full_name: Orientation.oangle_add_oangle_rev
code: /-- Adding the angles between two vectors in each order results in 0. -/
@[simp]
theorem oangle_add_oangle_rev (x y : V) : o.oangle x y + o.oangle y x = 0 
goal: o.oangle x y + o.oangle y x = 0

full_name: Orientation.oangle_neg_left
code: /-- Negating the first vector passed to `oangle` adds `π` to the angle. -/
theorem oangle_neg_left {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle (-x) y = o.oangle x y + π 
goal: o.oangle (-x) y = o.oangle x y + ↑π

full_name: Orientation.oangle_neg_right
code: /-- Negating the second vector passed to `oangle` adds `π` to the angle. -/
theorem oangle_neg_right {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle x (-y) = o.oangle x y + π 
goal: o.oangle x (-y) = o.oangle x y + ↑π

full_name: Orientation.two_zsmul_oangle_neg_left
code: /-- Negating the first vector passed to `oangle` does not change twice the angle. -/
@[simp]
theorem two_zsmul_oangle_neg_left (x y : V) :
    (2 : ℤ) • o.oangle (-x) y = (2 : ℤ) • o.oangle x y 
goal: 2 • o.oangle (-x) y = 2 • o.oangle x y

full_name: Orientation.two_zsmul_oangle_neg_right
code: /-- Negating the second vector passed to `oangle` does not change twice the angle. -/
@[simp]
theorem two_zsmul_oangle_neg_right (x y : V) :
    (2 : ℤ) • o.oangle x (-y) = (2 : ℤ) • o.oangle x y 
goal: 2 • o.oangle x (-y) = 2 • o.oangle x y

full_name: Orientation.oangle_neg_neg
code: /-- Negating both vectors passed to `oangle` does not change the angle. -/
@[simp]
theorem oangle_neg_neg (x y : V) : o.oangle (-x) (-y) = o.oangle x y 
goal: o.oangle (-x) (-y) = o.oangle x y

full_name: Orientation.oangle_neg_left_eq_neg_right
code: /-- Negating the first vector produces the same angle as negating the second vector. -/
theorem oangle_neg_left_eq_neg_right (x y : V) : o.oangle (-x) y = o.oangle x (-y) 
goal: o.oangle (-x) y = o.oangle x (-y)

full_name: Orientation.oangle_neg_self_left
code: /-- The angle between the negation of a nonzero vector and that vector is `π`. -/
@[simp]
theorem oangle_neg_self_left {x : V} (hx : x ≠ 0) : o.oangle (-x) x = π 
goal: o.oangle (-x) x = ↑π

full_name: Orientation.oangle_neg_self_right
code: /-- The angle between a nonzero vector and its negation is `π`. -/
@[simp]
theorem oangle_neg_self_right {x : V} (hx : x ≠ 0) : o.oangle x (-x) = π 
goal: o.oangle x (-x) = ↑π

full_name: Orientation.two_zsmul_oangle_neg_self_left
code: /-- Twice the angle between the negation of a vector and that vector is 0. -/
-- @[simp] -- Porting note (#10618): simp can prove this
theorem two_zsmul_oangle_neg_self_left (x : V) : (2 : ℤ) • o.oangle (-x) x = 0 
goal: 2 • o.oangle (-x) x = 0

full_name: Orientation.two_zsmul_oangle_neg_self_right
code: /-- Twice the angle between a vector and its negation is 0. -/
-- @[simp] -- Porting note (#10618): simp can prove this
theorem two_zsmul_oangle_neg_self_right (x : V) : (2 : ℤ) • o.oangle x (-x) = 0 
goal: 2 • o.oangle x (-x) = 0

full_name: Orientation.oangle_add_oangle_rev_neg_left
code: /-- Adding the angles between two vectors in each order, with the first vector in each angle
negated, results in 0. -/
@[simp]
theorem oangle_add_oangle_rev_neg_left (x y : V) : o.oangle (-x) y + o.oangle (-y) x = 0 
goal: o.oangle (-x) y + o.oangle (-y) x = 0

full_name: Orientation.oangle_add_oangle_rev_neg_right
code: /-- Adding the angles between two vectors in each order, with the second vector in each angle
negated, results in 0. -/
@[simp]
theorem oangle_add_oangle_rev_neg_right (x y : V) : o.oangle x (-y) + o.oangle y (-x) = 0 
goal: o.oangle x (-y) + o.oangle y (-x) = 0

full_name: Orientation.oangle_smul_left_of_pos
code: /-- Multiplying the first vector passed to `oangle` by a positive real does not change the
angle. -/
@[simp]
theorem oangle_smul_left_of_pos (x y : V) {r : ℝ} (hr : 0 < r) :
    o.oangle (r • x) y = o.oangle x y 
goal: o.oangle (r • x) y = o.oangle x y

full_name: Orientation.oangle_smul_right_of_pos
code: /-- Multiplying the second vector passed to `oangle` by a positive real does not change the
angle. -/
@[simp]
theorem oangle_smul_right_of_pos (x y : V) {r : ℝ} (hr : 0 < r) :
    o.oangle x (r • y) = o.oangle x y 
goal: o.oangle x (r • y) = o.oangle x y

full_name: Orientation.oangle_smul_left_of_neg
code: /-- Multiplying the first vector passed to `oangle` by a negative real produces the same angle
as negating that vector. -/
@[simp]
theorem oangle_smul_left_of_neg (x y : V) {r : ℝ} (hr : r < 0) :
    o.oangle (r • x) y = o.oangle (-x) y 
goal: o.oangle (r • x) y = o.oangle (-x) y

full_name: Orientation.oangle_smul_right_of_neg
code: /-- Multiplying the second vector passed to `oangle` by a negative real produces the same angle
as negating that vector. -/
@[simp]
theorem oangle_smul_right_of_neg (x y : V) {r : ℝ} (hr : r < 0) :
    o.oangle x (r • y) = o.oangle x (-y) 
goal: o.oangle x (r • y) = o.oangle x (-y)

full_name: Orientation.oangle_smul_left_self_of_nonneg
code: /-- The angle between a nonnegative multiple of a vector and that vector is 0. -/
@[simp]
theorem oangle_smul_left_self_of_nonneg (x : V) {r : ℝ} (hr : 0 ≤ r) : o.oangle (r • x) x = 0 
goal: o.oangle (r • x) x = 0

full_name: Orientation.oangle_smul_right_self_of_nonneg
code: /-- The angle between a vector and a nonnegative multiple of that vector is 0. -/
@[simp]
theorem oangle_smul_right_self_of_nonneg (x : V) {r : ℝ} (hr : 0 ≤ r) : o.oangle x (r • x) = 0 
goal: o.oangle x (r • x) = 0

full_name: Orientation.oangle_smul_smul_self_of_nonneg
code: /-- The angle between two nonnegative multiples of the same vector is 0. -/
@[simp]
theorem oangle_smul_smul_self_of_nonneg (x : V) {r₁ r₂ : ℝ} (hr₁ : 0 ≤ r₁) (hr₂ : 0 ≤ r₂) :
    o.oangle (r₁ • x) (r₂ • x) = 0 
goal: o.oangle (r₁ • x) (r₂ • x) = 0

full_name: Orientation.two_zsmul_oangle_smul_left_of_ne_zero
code: /-- Multiplying the first vector passed to `oangle` by a nonzero real does not change twice the
angle. -/
@[simp]
theorem two_zsmul_oangle_smul_left_of_ne_zero (x y : V) {r : ℝ} (hr : r ≠ 0) :
    (2 : ℤ) • o.oangle (r • x) y = (2 : ℤ) • o.oangle x y 
goal: 2 • o.oangle (r • x) y = 2 • o.oangle x y

full_name: Orientation.two_zsmul_oangle_smul_right_of_ne_zero
code: /-- Multiplying the second vector passed to `oangle` by a nonzero real does not change twice the
angle. -/
@[simp]
theorem two_zsmul_oangle_smul_right_of_ne_zero (x y : V) {r : ℝ} (hr : r ≠ 0) :
    (2 : ℤ) • o.oangle x (r • y) = (2 : ℤ) • o.oangle x y 
goal: 2 • o.oangle x (r • y) = 2 • o.oangle x y

full_name: Orientation.two_zsmul_oangle_smul_left_self
code: /-- Twice the angle between a multiple of a vector and that vector is 0. -/
@[simp]
theorem two_zsmul_oangle_smul_left_self (x : V) {r : ℝ} : (2 : ℤ) • o.oangle (r • x) x = 0 
goal: 2 • o.oangle (r • x) x = 0

full_name: Orientation.two_zsmul_oangle_smul_right_self
code: /-- Twice the angle between a vector and a multiple of that vector is 0. -/
@[simp]
theorem two_zsmul_oangle_smul_right_self (x : V) {r : ℝ} : (2 : ℤ) • o.oangle x (r • x) = 0 
goal: 2 • o.oangle x (r • x) = 0

full_name: Orientation.two_zsmul_oangle_smul_smul_self
code: /-- Twice the angle between two multiples of a vector is 0. -/
@[simp]
theorem two_zsmul_oangle_smul_smul_self (x : V) {r₁ r₂ : ℝ} :
    (2 : ℤ) • o.oangle (r₁ • x) (r₂ • x) = 0 
goal: 2 • o.oangle (r₁ • x) (r₂ • x) = 0

full_name: Orientation.two_zsmul_oangle_left_of_span_eq
code: /-- If the spans of two vectors are equal, twice angles with those vectors on the left are
equal. -/
theorem two_zsmul_oangle_left_of_span_eq {x y : V} (z : V) (h : (ℝ ∙ x) = ℝ ∙ y) :
    (2 : ℤ) • o.oangle x z = (2 : ℤ) • o.oangle y z 
goal: 2 • o.oangle x z = 2 • o.oangle y z

full_name: Orientation.two_zsmul_oangle_right_of_span_eq
code: /-- If the spans of two vectors are equal, twice angles with those vectors on the right are
equal. -/
theorem two_zsmul_oangle_right_of_span_eq (x : V) {y z : V} (h : (ℝ ∙ y) = ℝ ∙ z) :
    (2 : ℤ) • o.oangle x y = (2 : ℤ) • o.oangle x z 
goal: 2 • o.oangle x y = 2 • o.oangle x z

full_name: Orientation.two_zsmul_oangle_of_span_eq_of_span_eq
code: /-- If the spans of two pairs of vectors are equal, twice angles between those vectors are
equal. -/
theorem two_zsmul_oangle_of_span_eq_of_span_eq {w x y z : V} (hwx : (ℝ ∙ w) = ℝ ∙ x)
    (hyz : (ℝ ∙ y) = ℝ ∙ z) : (2 : ℤ) • o.oangle w y = (2 : ℤ) • o.oangle x z 
goal: 2 • o.oangle w y = 2 • o.oangle x z

full_name: Orientation.oangle_eq_zero_iff_oangle_rev_eq_zero
code: /-- The oriented angle between two vectors is zero if and only if the angle with the vectors
swapped is zero. -/
theorem oangle_eq_zero_iff_oangle_rev_eq_zero {x y : V} : o.oangle x y = 0 ↔ o.oangle y x = 0 
goal: o.oangle x y = 0 ↔ o.oangle y x = 0

full_name: Orientation.oangle_eq_zero_iff_sameRay
code: /-- The oriented angle between two vectors is zero if and only if they are on the same ray. -/
theorem oangle_eq_zero_iff_sameRay {x y : V} : o.oangle x y = 0 ↔ SameRay ℝ x y 
goal: o.oangle x y = 0 ↔ SameRay ℝ x y

full_name: Orientation.oangle_eq_pi_iff_oangle_rev_eq_pi
code: /-- The oriented angle between two vectors is `π` if and only if the angle with the vectors
swapped is `π`. -/
theorem oangle_eq_pi_iff_oangle_rev_eq_pi {x y : V} : o.oangle x y = π ↔ o.oangle y x = π 
goal: o.oangle x y = ↑π ↔ o.oangle y x = ↑π

full_name: Orientation.oangle_eq_pi_iff_sameRay_neg
code: /-- The oriented angle between two vectors is `π` if and only they are nonzero and the first is
on the same ray as the negation of the second. -/
theorem oangle_eq_pi_iff_sameRay_neg {x y : V} :
    o.oangle x y = π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y) 
goal: o.oangle x y = ↑π ↔ x ≠ 0 ∧ y ≠ 0 ∧ SameRay ℝ x (-y)

full_name: Orientation.oangle_eq_zero_or_eq_pi_iff_not_linearIndependent
code: /-- The oriented angle between two vectors is zero or `π` if and only if those two vectors are
not linearly independent. -/
theorem oangle_eq_zero_or_eq_pi_iff_not_linearIndependent {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ ¬LinearIndependent ℝ ![x, y] 
goal: o.oangle x y = 0 ∨ o.oangle x y = ↑π ↔ ¬LinearIndependent ℝ ![x, y]

full_name: Orientation.oangle_eq_zero_or_eq_pi_iff_right_eq_smul
code: /-- The oriented angle between two vectors is zero or `π` if and only if the first vector is zero
or the second is a multiple of the first. -/
theorem oangle_eq_zero_or_eq_pi_iff_right_eq_smul {x y : V} :
    o.oangle x y = 0 ∨ o.oangle x y = π ↔ x = 0 ∨ ∃ r : ℝ, y = r • x 
goal: o.oangle x y = 0 ∨ o.oangle x y = ↑π ↔ x = 0 ∨ ∃ r, y = r • x

full_name: Orientation.oangle_ne_zero_and_ne_pi_iff_linearIndependent
code: /-- The oriented angle between two vectors is not zero or `π` if and only if those two vectors
are linearly independent. -/
theorem oangle_ne_zero_and_ne_pi_iff_linearIndependent {x y : V} :
    o.oangle x y ≠ 0 ∧ o.oangle x y ≠ π ↔ LinearIndependent ℝ ![x, y] 
goal: o.oangle x y ≠ 0 ∧ o.oangle x y ≠ ↑π ↔ LinearIndependent ℝ ![x, y]

full_name: Orientation.eq_iff_norm_eq_and_oangle_eq_zero
code: /-- Two vectors are equal if and only if they have equal norms and zero angle between them. -/
theorem eq_iff_norm_eq_and_oangle_eq_zero (x y : V) : x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0 
goal: x = y ↔ ‖x‖ = ‖y‖ ∧ o.oangle x y = 0

full_name: Orientation.eq_iff_oangle_eq_zero_of_norm_eq
code: /-- Two vectors with equal norms are equal if and only if they have zero angle between them. -/
theorem eq_iff_oangle_eq_zero_of_norm_eq {x y : V} (h : ‖x‖ = ‖y‖) : x = y ↔ o.oangle x y = 0 
goal: x = y ↔ o.oangle x y = 0

full_name: Orientation.eq_iff_norm_eq_of_oangle_eq_zero
code: /-- Two vectors with zero angle between them are equal if and only if they have equal norms. -/
theorem eq_iff_norm_eq_of_oangle_eq_zero {x y : V} (h : o.oangle x y = 0) : x = y ↔ ‖x‖ = ‖y‖ 
goal: x = y ↔ ‖x‖ = ‖y‖

full_name: Orientation.oangle_add
code: /-- Given three nonzero vectors, the angle between the first and the second plus the angle
between the second and the third equals the angle between the first and the third. -/
@[simp]
theorem oangle_add {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z = o.oangle x z 
goal: o.oangle x y + o.oangle y z = o.oangle x z

full_name: Orientation.oangle_add_swap
code: /-- Given three nonzero vectors, the angle between the second and the third plus the angle
between the first and the second equals the angle between the first and the third. -/
@[simp]
theorem oangle_add_swap {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle y z + o.oangle x y = o.oangle x z 
goal: o.oangle y z + o.oangle x y = o.oangle x z

full_name: Orientation.oangle_sub_left
code: /-- Given three nonzero vectors, the angle between the first and the third minus the angle
between the first and the second equals the angle between the second and the third. -/
@[simp]
theorem oangle_sub_left {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x z - o.oangle x y = o.oangle y z 
goal: o.oangle x z - o.oangle x y = o.oangle y z

full_name: Orientation.oangle_sub_right
code: /-- Given three nonzero vectors, the angle between the first and the third minus the angle
between the second and the third equals the angle between the first and the second. -/
@[simp]
theorem oangle_sub_right {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x z - o.oangle y z = o.oangle x y 
goal: o.oangle x z - o.oangle y z = o.oangle x y

full_name: Orientation.oangle_add_cyc3
code: /-- Given three nonzero vectors, adding the angles between them in cyclic order results in 0. -/
@[simp]
theorem oangle_add_cyc3 {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x y + o.oangle y z + o.oangle z x = 0 
goal: o.oangle x y + o.oangle y z + o.oangle z x = 0

full_name: Orientation.oangle_add_cyc3_neg_left
code: /-- Given three nonzero vectors, adding the angles between them in cyclic order, with the first
vector in each angle negated, results in π. If the vectors add to 0, this is a version of the
sum of the angles of a triangle. -/
@[simp]
theorem oangle_add_cyc3_neg_left {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle (-x) y + o.oangle (-y) z + o.oangle (-z) x = π 
goal: o.oangle (-x) y + o.oangle (-y) z + o.oangle (-z) x = ↑π

full_name: Orientation.oangle_add_cyc3_neg_right
code: /-- Given three nonzero vectors, adding the angles between them in cyclic order, with the second
vector in each angle negated, results in π. If the vectors add to 0, this is a version of the
sum of the angles of a triangle. -/
@[simp]
theorem oangle_add_cyc3_neg_right {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
    o.oangle x (-y) + o.oangle y (-z) + o.oangle z (-x) = π 
goal: o.oangle x (-y) + o.oangle y (-z) + o.oangle z (-x) = ↑π

full_name: Orientation.oangle_sub_eq_oangle_sub_rev_of_norm_eq
code: /-- Pons asinorum, oriented vector angle form. -/
theorem oangle_sub_eq_oangle_sub_rev_of_norm_eq {x y : V} (h : ‖x‖ = ‖y‖) :
    o.oangle x (x - y) = o.oangle (y - x) y 
goal: o.oangle x (x - y) = o.oangle (y - x) y

full_name: Orientation.oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq
code: /-- The angle at the apex of an isosceles triangle is `π` minus twice a base angle, oriented
vector angle form. -/
theorem oangle_eq_pi_sub_two_zsmul_oangle_sub_of_norm_eq {x y : V} (hn : x ≠ y) (h : ‖x‖ = ‖y‖) :
    o.oangle y x = π - (2 : ℤ) • o.oangle (y - x) y 
goal: o.oangle y x = ↑π - 2 • o.oangle (y - x) y

full_name: Orientation.oangle_map
code: /-- The angle between two vectors, with respect to an orientation given by `Orientation.map`
with a linear isometric equivalence, equals the angle between those two vectors, transformed by
the inverse of that equivalence, with respect to the original orientation. -/
@[simp]
theorem oangle_map (x y : V') (f : V ≃ₗᵢ[ℝ] V') :
    (Orientation.map (Fin 2) f.toLinearEquiv o).oangle x y = o.oangle (f.symm x) (f.symm y) 
goal: ((Orientation.map (Fin 2) f.toLinearEquiv) o).oangle x y = o.oangle (f.symm x) (f.symm y)

full_name: Complex.oangle
code: @[simp]
protected theorem _root_.Complex.oangle (w z : ℂ) :
    Complex.orientation.oangle w z = Complex.arg (conj w * z) 
goal: Complex.orientation.oangle w z = ↑((starRingEnd ℂ) w * z).arg

full_name: Orientation.oangle_map_complex
code: /-- The oriented angle on an oriented real inner product space of dimension 2 can be evaluated in
terms of a complex-number representation of the space. -/
theorem oangle_map_complex (f : V ≃ₗᵢ[ℝ] ℂ)
    (hf : Orientation.map (Fin 2) f.toLinearEquiv o = Complex.orientation) (x y : V) :
    o.oangle x y = Complex.arg (conj (f x) * f y) 
goal: o.oangle x y = ↑((starRingEnd ℂ) (f x) * f y).arg

full_name: Orientation.oangle_neg_orientation_eq_neg
code: /-- Negating the orientation negates the value of `oangle`. -/
theorem oangle_neg_orientation_eq_neg (x y : V) : (-o).oangle x y = -o.oangle x y 
goal: (-o).oangle x y = -o.oangle x y

full_name: Orientation.inner_eq_norm_mul_norm_mul_cos_oangle
code: /-- The inner product of two vectors is the product of the norms and the cosine of the oriented
angle between the vectors. -/
theorem inner_eq_norm_mul_norm_mul_cos_oangle (x y : V) :
    ⟪x, y⟫ = ‖x‖ * ‖y‖ * Real.Angle.cos (o.oangle x y) 
goal: ⟪x, y⟫_ℝ = ‖x‖ * ‖y‖ * (o.oangle x y).cos

full_name: Orientation.cos_oangle_eq_inner_div_norm_mul_norm
code: /-- The cosine of the oriented angle between two nonzero vectors is the inner product divided by
the product of the norms. -/
theorem cos_oangle_eq_inner_div_norm_mul_norm {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.Angle.cos (o.oangle x y) = ⟪x, y⟫ / (‖x‖ * ‖y‖) 
goal: (o.oangle x y).cos = ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖)

full_name: Orientation.cos_oangle_eq_cos_angle
code: /-- The cosine of the oriented angle between two nonzero vectors equals that of the unoriented
angle. -/
theorem cos_oangle_eq_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    Real.Angle.cos (o.oangle x y) = Real.cos (InnerProductGeometry.angle x y) 
goal: (o.oangle x y).cos = Real.cos (InnerProductGeometry.angle x y)

full_name: Orientation.oangle_eq_angle_or_eq_neg_angle
code: /-- The oriented angle between two nonzero vectors is plus or minus the unoriented angle. -/
theorem oangle_eq_angle_or_eq_neg_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle x y = InnerProductGeometry.angle x y ∨
      o.oangle x y = -InnerProductGeometry.angle x y 
goal: o.oangle x y = ↑(InnerProductGeometry.angle x y) ∨ o.oangle x y = -↑(InnerProductGeometry.angle x y)

full_name: Orientation.angle_eq_abs_oangle_toReal
code: /-- The unoriented angle between two nonzero vectors is the absolute value of the oriented angle,
converted to a real. -/
theorem angle_eq_abs_oangle_toReal {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    InnerProductGeometry.angle x y = |(o.oangle x y).toReal| 
goal: InnerProductGeometry.angle x y = |(o.oangle x y).toReal|

full_name: Orientation.eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero
code: /-- If the sign of the oriented angle between two vectors is zero, either one of the vectors is
zero or the unoriented angle is 0 or π. -/
theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {x y : V}
    (h : (o.oangle x y).sign = 0) :
    x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π 
goal: x = 0 ∨ y = 0 ∨ InnerProductGeometry.angle x y = 0 ∨ InnerProductGeometry.angle x y = π

full_name: Orientation.oangle_eq_of_angle_eq_of_sign_eq
code: /-- If two unoriented angles are equal, and the signs of the corresponding oriented angles are
equal, then the oriented angles are equal (even in degenerate cases). -/
theorem oangle_eq_of_angle_eq_of_sign_eq {w x y z : V}
    (h : InnerProductGeometry.angle w x = InnerProductGeometry.angle y z)
    (hs : (o.oangle w x).sign = (o.oangle y z).sign) : o.oangle w x = o.oangle y z 
goal: o.oangle w x = o.oangle y z

full_name: Orientation.angle_eq_iff_oangle_eq_of_sign_eq
code: /-- If the signs of two oriented angles between nonzero vectors are equal, the oriented angles are
equal if and only if the unoriented angles are equal. -/
theorem angle_eq_iff_oangle_eq_of_sign_eq {w x y z : V} (hw : w ≠ 0) (hx : x ≠ 0) (hy : y ≠ 0)
    (hz : z ≠ 0) (hs : (o.oangle w x).sign = (o.oangle y z).sign) :
    InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔
    o.oangle w x = o.oangle y z 
goal: InnerProductGeometry.angle w x = InnerProductGeometry.angle y z ↔ o.oangle w x = o.oangle y z

full_name: Orientation.oangle_eq_angle_of_sign_eq_one
code: /-- The oriented angle between two vectors equals the unoriented angle if the sign is positive. -/
theorem oangle_eq_angle_of_sign_eq_one {x y : V} (h : (o.oangle x y).sign = 1) :
    o.oangle x y = InnerProductGeometry.angle x y 
goal: o.oangle x y = ↑(InnerProductGeometry.angle x y)

full_name: Orientation.oangle_eq_neg_angle_of_sign_eq_neg_one
code: /-- The oriented angle between two vectors equals minus the unoriented angle if the sign is
negative. -/
theorem oangle_eq_neg_angle_of_sign_eq_neg_one {x y : V} (h : (o.oangle x y).sign = -1) :
    o.oangle x y = -InnerProductGeometry.angle x y 
goal: o.oangle x y = -↑(InnerProductGeometry.angle x y)

full_name: Orientation.oangle_eq_zero_iff_angle_eq_zero
code: /-- The oriented angle between two nonzero vectors is zero if and only if the unoriented angle
is zero. -/
theorem oangle_eq_zero_iff_angle_eq_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
    o.oangle x y = 0 ↔ InnerProductGeometry.angle x y = 0 
goal: o.oangle x y = 0 ↔ InnerProductGeometry.angle x y = 0

full_name: Orientation.oangle_eq_pi_iff_angle_eq_pi
code: /-- The oriented angle between two vectors is `π` if and only if the unoriented angle is `π`. -/
theorem oangle_eq_pi_iff_angle_eq_pi {x y : V} :
    o.oangle x y = π ↔ InnerProductGeometry.angle x y = π 
goal: o.oangle x y = ↑π ↔ InnerProductGeometry.angle x y = π

full_name: Orientation.eq_zero_or_oangle_eq_iff_inner_eq_zero
code: /-- One of two vectors is zero or the oriented angle between them is plus or minus `π / 2` if
and only if the inner product of those vectors is zero. -/
theorem eq_zero_or_oangle_eq_iff_inner_eq_zero {x y : V} :
    x = 0 ∨ y = 0 ∨ o.oangle x y = (π / 2 : ℝ) ∨ o.oangle x y = (-π / 2 : ℝ) ↔ ⟪x, y⟫ = 0 
goal: x = 0 ∨ y = 0 ∨ o.oangle x y = ↑(π / 2) ∨ o.oangle x y = ↑(-π / 2) ↔ ⟪x, y⟫_ℝ = 0

full_name: Orientation.inner_eq_zero_of_oangle_eq_pi_div_two
code: /-- If the oriented angle between two vectors is `π / 2`, the inner product of those vectors
is zero. -/
theorem inner_eq_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) :
    ⟪x, y⟫ = 0 
goal: ⟪x, y⟫_ℝ = 0

full_name: Orientation.inner_rev_eq_zero_of_oangle_eq_pi_div_two
code: /-- If the oriented angle between two vectors is `π / 2`, the inner product of those vectors
(reversed) is zero. -/
theorem inner_rev_eq_zero_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = (π / 2 : ℝ)) :
    ⟪y, x⟫ = 0 
goal: ⟪y, x⟫_ℝ = 0

full_name: Orientation.inner_eq_zero_of_oangle_eq_neg_pi_div_two
code: /-- If the oriented angle between two vectors is `-π / 2`, the inner product of those vectors
is zero. -/
theorem inner_eq_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    ⟪x, y⟫ = 0 
goal: ⟪x, y⟫_ℝ = 0

full_name: Orientation.inner_rev_eq_zero_of_oangle_eq_neg_pi_div_two
code: /-- If the oriented angle between two vectors is `-π / 2`, the inner product of those vectors
(reversed) is zero. -/
theorem inner_rev_eq_zero_of_oangle_eq_neg_pi_div_two {x y : V} (h : o.oangle x y = (-π / 2 : ℝ)) :
    ⟪y, x⟫ = 0 
goal: ⟪y, x⟫_ℝ = 0

full_name: Orientation.oangle_sign_neg_left
code: /-- Negating the first vector passed to `oangle` negates the sign of the angle. -/
@[simp]
theorem oangle_sign_neg_left (x y : V) : (o.oangle (-x) y).sign = -(o.oangle x y).sign 
goal: (o.oangle (-x) y).sign = -(o.oangle x y).sign

full_name: Orientation.oangle_sign_neg_right
code: /-- Negating the second vector passed to `oangle` negates the sign of the angle. -/
@[simp]
theorem oangle_sign_neg_right (x y : V) : (o.oangle x (-y)).sign = -(o.oangle x y).sign 
goal: (o.oangle x (-y)).sign = -(o.oangle x y).sign

full_name: Orientation.oangle_sign_smul_left
code: /-- Multiplying the first vector passed to `oangle` by a real multiplies the sign of the angle by
the sign of the real. -/
@[simp]
theorem oangle_sign_smul_left (x y : V) (r : ℝ) :
    (o.oangle (r • x) y).sign = SignType.sign r * (o.oangle x y).sign 
goal: (o.oangle (r • x) y).sign = SignType.sign r * (o.oangle x y).sign

full_name: Orientation.oangle_sign_smul_right
code: /-- Multiplying the second vector passed to `oangle` by a real multiplies the sign of the angle by
the sign of the real. -/
@[simp]
theorem oangle_sign_smul_right (x y : V) (r : ℝ) :
    (o.oangle x (r • y)).sign = SignType.sign r * (o.oangle x y).sign 
goal: (o.oangle x (r • y)).sign = SignType.sign r * (o.oangle x y).sign

full_name: Orientation.oangle_smul_add_right_eq_zero_or_eq_pi_iff
code: /-- Auxiliary lemma for the proof of `oangle_sign_smul_add_right`; not intended to be used
outside of that proof. -/
theorem oangle_smul_add_right_eq_zero_or_eq_pi_iff {x y : V} (r : ℝ) :
    o.oangle x (r • x + y) = 0 ∨ o.oangle x (r • x + y) = π ↔
    o.oangle x y = 0 ∨ o.oangle x y = π 
goal: o.oangle x (r • x + y) = 0 ∨ o.oangle x (r • x + y) = ↑π ↔ o.oangle x y = 0 ∨ o.oangle x y = ↑π

full_name: Orientation.oangle_sign_smul_add_right
code: /-- Adding a multiple of the first vector passed to `oangle` to the second vector does not change
the sign of the angle. -/
@[simp]
theorem oangle_sign_smul_add_right (x y : V) (r : ℝ) :
    (o.oangle x (r • x + y)).sign = (o.oangle x y).sign 
goal: (o.oangle x (r • x + y)).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_add_smul_left
code: /-- Adding a multiple of the second vector passed to `oangle` to the first vector does not change
the sign of the angle. -/
@[simp]
theorem oangle_sign_add_smul_left (x y : V) (r : ℝ) :
    (o.oangle (x + r • y) y).sign = (o.oangle x y).sign 
goal: (o.oangle (x + r • y) y).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_smul_right
code: /-- Subtracting a multiple of the first vector passed to `oangle` from the second vector does
not change the sign of the angle. -/
@[simp]
theorem oangle_sign_sub_smul_right (x y : V) (r : ℝ) :
    (o.oangle x (y - r • x)).sign = (o.oangle x y).sign 
goal: (o.oangle x (y - r • x)).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_smul_left
code: /-- Subtracting a multiple of the second vector passed to `oangle` from the first vector does
not change the sign of the angle. -/
@[simp]
theorem oangle_sign_sub_smul_left (x y : V) (r : ℝ) :
    (o.oangle (x - r • y) y).sign = (o.oangle x y).sign 
goal: (o.oangle (x - r • y) y).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_add_right
code: /-- Adding the first vector passed to `oangle` to the second vector does not change the sign of
the angle. -/
@[simp]
theorem oangle_sign_add_right (x y : V) : (o.oangle x (x + y)).sign = (o.oangle x y).sign 
goal: (o.oangle x (x + y)).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_add_left
code: /-- Adding the second vector passed to `oangle` to the first vector does not change the sign of
the angle. -/
@[simp]
theorem oangle_sign_add_left (x y : V) : (o.oangle (x + y) y).sign = (o.oangle x y).sign 
goal: (o.oangle (x + y) y).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_right
code: /-- Subtracting the first vector passed to `oangle` from the second vector does not change the
sign of the angle. -/
@[simp]
theorem oangle_sign_sub_right (x y : V) : (o.oangle x (y - x)).sign = (o.oangle x y).sign 
goal: (o.oangle x (y - x)).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_left
code: /-- Subtracting the second vector passed to `oangle` from the first vector does not change the
sign of the angle. -/
@[simp]
theorem oangle_sign_sub_left (x y : V) : (o.oangle (x - y) y).sign = (o.oangle x y).sign 
goal: (o.oangle (x - y) y).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_smul_sub_right
code: /-- Subtracting the second vector passed to `oangle` from a multiple of the first vector negates
the sign of the angle. -/
@[simp]
theorem oangle_sign_smul_sub_right (x y : V) (r : ℝ) :
    (o.oangle x (r • x - y)).sign = -(o.oangle x y).sign 
goal: (o.oangle x (r • x - y)).sign = -(o.oangle x y).sign

full_name: Orientation.oangle_sign_smul_sub_left
code: /-- Subtracting the first vector passed to `oangle` from a multiple of the second vector negates
the sign of the angle. -/
@[simp]
theorem oangle_sign_smul_sub_left (x y : V) (r : ℝ) :
    (o.oangle (r • y - x) y).sign = -(o.oangle x y).sign 
goal: (o.oangle (r • y - x) y).sign = -(o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_right_eq_neg
code: /-- Subtracting the second vector passed to `oangle` from the first vector negates the sign of
the angle. -/
theorem oangle_sign_sub_right_eq_neg (x y : V) :
    (o.oangle x (x - y)).sign = -(o.oangle x y).sign 
goal: (o.oangle x (x - y)).sign = -(o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_left_eq_neg
code: /-- Subtracting the first vector passed to `oangle` from the second vector negates the sign of
the angle. -/
theorem oangle_sign_sub_left_eq_neg (x y : V) :
    (o.oangle (y - x) y).sign = -(o.oangle x y).sign 
goal: (o.oangle (y - x) y).sign = -(o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_right_swap
code: /-- Subtracting the first vector passed to `oangle` from the second vector then swapping the
vectors does not change the sign of the angle. -/
@[simp]
theorem oangle_sign_sub_right_swap (x y : V) : (o.oangle y (y - x)).sign = (o.oangle x y).sign 
goal: (o.oangle y (y - x)).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_sub_left_swap
code: /-- Subtracting the second vector passed to `oangle` from the first vector then swapping the
vectors does not change the sign of the angle. -/
@[simp]
theorem oangle_sign_sub_left_swap (x y : V) : (o.oangle (x - y) x).sign = (o.oangle x y).sign 
goal: (o.oangle (x - y) x).sign = (o.oangle x y).sign

full_name: Orientation.oangle_sign_smul_add_smul_right
code: /-- The sign of the angle between a vector, and a linear combination of that vector with a second
vector, is the sign of the factor by which the second vector is multiplied in that combination
multiplied by the sign of the angle between the two vectors. -/
-- @[simp] -- Porting note (#10618): simp can prove this
theorem oangle_sign_smul_add_smul_right (x y : V) (r₁ r₂ : ℝ) :
    (o.oangle x (r₁ • x + r₂ • y)).sign = SignType.sign r₂ * (o.oangle x y).sign 
goal: (o.oangle x (r₁ • x + r₂ • y)).sign = SignType.sign r₂ * (o.oangle x y).sign

full_name: Orientation.oangle_sign_smul_add_smul_left
code: /-- The sign of the angle between a linear combination of two vectors and the second vector is
the sign of the factor by which the first vector is multiplied in that combination multiplied by
the sign of the angle between the two vectors. -/
-- @[simp] -- Porting note (#10618): simp can prove this
theorem oangle_sign_smul_add_smul_left (x y : V) (r₁ r₂ : ℝ) :
    (o.oangle (r₁ • x + r₂ • y) y).sign = SignType.sign r₁ * (o.oangle x y).sign 
goal: (o.oangle (r₁ • x + r₂ • y) y).sign = SignType.sign r₁ * (o.oangle x y).sign

full_name: Orientation.oangle_sign_smul_add_smul_smul_add_smul
code: /-- The sign of the angle between two linear combinations of two vectors is the sign of the
determinant of the factors in those combinations multiplied by the sign of the angle between the
two vectors. -/
theorem oangle_sign_smul_add_smul_smul_add_smul (x y : V) (r₁ r₂ r₃ r₄ : ℝ) :
    (o.oangle (r₁ • x + r₂ • y) (r₃ • x + r₄ • y)).sign =
      SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign 
goal: (o.oangle (r₁ • x + r₂ • y) (r₃ • x + r₄ • y)).sign = SignType.sign (r₁ * r₄ - r₂ * r₃) * (o.oangle x y).sign

full_name: Orientation.abs_oangle_sub_left_toReal_lt_pi_div_two
code: /-- A base angle of an isosceles triangle is acute, oriented vector angle form. -/
theorem abs_oangle_sub_left_toReal_lt_pi_div_two {x y : V} (h : ‖x‖ = ‖y‖) :
    |(o.oangle (y - x) y).toReal| < π / 2 
goal: |(o.oangle (y - x) y).toReal| < π / 2

full_name: Orientation.abs_oangle_sub_right_toReal_lt_pi_div_two
code: /-- A base angle of an isosceles triangle is acute, oriented vector angle form. -/
theorem abs_oangle_sub_right_toReal_lt_pi_div_two {x y : V} (h : ‖x‖ = ‖y‖) :
    |(o.oangle x (x - y)).toReal| < π / 2 
goal: |(o.oangle x (x - y)).toReal| < π / 2