∀ {A : Type u_1} [inst : PartialOrder A] [inst_1 : Ring A] [inst_2 : StarRing A] [inst_3 : TopologicalSpace A]
  [inst_4 : StarOrderedRing A] [inst_5 : Algebra ℝ A] [inst_6 : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint]
  [inst_7 : NonnegSpectrumClass ℝ A] [IsSemitopologicalRing A] [T2Space A] {a : A},
  [IsStrictlyPositive a, IsStrictlyPositive (CFC.sqrt a) ∧ a = CFC.sqrt a * CFC.sqrt a,
      IsUnit (CFC.sqrt a) ∧ a = CFC.sqrt a * CFC.sqrt a, ∃ b, IsStrictlyPositive b ∧ a = b * b,
      ∃ b, IsUnit b ∧ IsSelfAdjoint b ∧ a = b * b, ∃ b, IsUnit b ∧ a = star b * b, ∃ b, IsUnit b ∧ a = b * star b,
      0 ≤ a ∧ IsUnit a, IsSelfAdjoint a ∧ ∀ x ∈ spectrum ℝ a, 0 < x].TFAE