∀ {R : Type u_1} [inst : Ring R] [inst_1 : LinearOrder R] {η : R} {P : Polynomial R} (d : ℕ),
  (∀ m < d + 1,
      ∀ {P : Polynomial R},
        P ≠ 0 → P.natDegree = m → P.signVariations + 1 ≤ ((Polynomial.X - Polynomial.C η) * P).signVariations) →
    P.eraseLead.natDegree < d + 1 →
      ¬P.eraseLead = 0 →
        ((Polynomial.X - Polynomial.C η) * P).eraseLead.signVariations + 1 =
            ((Polynomial.X - Polynomial.C η) * P).signVariations →
          ((Polynomial.X - Polynomial.C η) * P.eraseLead).signVariations ≤
              ((Polynomial.X - Polynomial.C η) * P).eraseLead.signVariations →
            P.signVariations + 1 ≤ ((Polynomial.X - Polynomial.C η) * P).signVariations