suffices : (pmag * Real.cos pangle)^2 + (pmag * Real.sin pangle)^2 = pmag^2
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
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
this : (pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2 = pmag ^ 2
⊢ Real.sqrt ((pmag * Real.cos pangle) ^ 2 + (pmag * Real.sin pangle) ^ 2) = pmag

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