English
In a nonunital nonassoc semiring, mulRight b = mulLeft b iff ∀ a, Commute a b.
Русский
В ненулевом неассоциативном полугруппе: mulRight b = mulLeft b тогда и только тогда, когда ∀ a, a и b коммутируют.
LaTeX
$$$mulRight\, b = mulLeft\, b \iff \forall a,\ Commute a b$$$
Lean4
/-- Vieta's formula for a quadratic equation, relating the coefficients of the polynomial with
its roots. This particular version states that if we have a root `x` of a monic quadratic
polynomial, then there is another root `y` such that `x + y` is negative the `a_1` coefficient
and `x * y` is the `a_0` coefficient. -/
theorem vieta_formula_quadratic {b c x : α} (h : x * x - b * x + c = 0) :
∃ y : α, y * y - b * y + c = 0 ∧ x + y = b ∧ x * y = c :=
by
have : c = x * (b - x) := (eq_neg_of_add_eq_zero_right h).trans (by simp [mul_sub, mul_comm])
refine ⟨b - x, ?_, by simp, by rw [this]⟩
rw [this, sub_add, ← sub_mul, sub_self]
