have h : (Real.sin pangle)^2 + (Real.cos pangle)^2 = 1 :=
  Real.sin_sq_add_cos_sq pangle
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
case this
pangle pmag : ℝ
hpangle1 : -(Real.pi / 2) < { angle := pangle, magnitude := pmag }.angle
hpangle2 : { angle := pangle, magnitude := pmag }.angle < Real.pi / 2
hpmag : 0 < pmag
h : Real.sin pangle ^ 2 + Real.cos pangle ^ 2 = 1
⊢ (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2