{ suffices : (pmag * Real.cos pangle)^2 + (pmag * Real.sin pangle)^2 = pmag^2
  rw [this]
  rw [Real.sqrt_sq] 
  linarith
  have h : (Real.sin pangle)^2 + (Real.cos pangle)^2 = 1 :=
    Real.sin_sq_add_cos_sq pangle
  linear_combination pmag ^ 2 * h }
case mk.right
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag
no goals