reserve z1,z2,z3,z4,z for Quaternion;

theorem
  z * (1/(|.z.| ^2) * z*') = [* (|.z.| ^2)/(|.z.| ^2),0,0,0*]
proof
  set zz = |.z.| ^2;
  set z3 = 1/zz * z*';
  z*(1/zz * z*')=
  (Rea z * Rea z3 - Im1 z * Im1 z3 - Im2 z * Im2 z3 - Im3 z * Im3 z3) +
  (Rea z * Im1 z3 + Im1 z * Rea z3 + Im2 z * Im3 z3 - Im3 z * Im2 z3)*<i> +
  (Rea z * Im2 z3 + Im2 z * Rea z3 + Im3 z * Im1 z3 - Im1 z * Im3 z3)*<j> +
  (Rea z * Im3 z3 + Im3 z * Rea z3 + Im1 z * Im2 z3 - Im2 z * Im1 z3)*<k>
    by QUATERNI:93
  .= [*(Rea z * Rea (z3) - Im1 z * Im1 (z3) - Im2 z * Im2 (z3
) - Im3 z * Im3 (z3)), (Rea z * Im1 (z3) + Im1 z * Rea (z3) + Im2 z * Im3 (z3)
- Im3 z * Im2 (z3)), (Rea z * Im2 (z3) + Im2 z * Rea (z3) + Im3 z * Im1 (z3) -
  Im1 z * Im3 (z3)), (Rea z * Im3 (z3) + Im3 z * Rea (z3) + Im1 z * Im2 (z3) -
  Im2 z * Im1 (z3))*] by QUATERN2:1
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * Im1 (z3) - Im2 z * Im2 (z3) - Im3 z
* Im3 (z3)), (Rea z * Im1 (z3) + Im1 z * Rea (z3) + Im2 z * Im3 (z3) - Im3 z *
  Im2 (z3)), (Rea z * Im2 (z3) + Im2 z * Rea (z3) + Im3 z * Im1 (z3) - Im1 z *
  Im3 (z3)), (Rea z * Im3 (z3) + Im3 z * Rea (z3) + Im1 z * Im2 (z3) - Im2 z *
  Im1 (z3))*] by Th24
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * Im2 (z3) -
Im3 z * Im3 (z3)), (Rea z * Im1 (z3)+ Im1 z * Rea (z3) + Im2 z * Im3 (z3) - Im3
z * Im2 (z3)), (Rea z * Im2 (z3) + Im2 z * Rea (z3) + Im3 z * Im1 (z3) - Im1 z
* Im3 (z3)), (Rea z * Im3 (z3) + Im3 z * Rea (z3) + Im1 z * Im2 (z3) - Im2 z *
  Im1 (z3))*] by Th25
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * Im3 (z3)), (Rea z * Im1 (z3)+ Im1 z * Rea (z3) + Im2 z * Im3 (z3) -
  Im3 z * Im2 (z3)), (Rea z * Im2 (z3) + Im2 z * Rea (z3) + Im3 z * Im1 (z3) -
  Im1 z * Im3 (z3)), (Rea z * Im3 (z3) + Im3 z * Rea (z3) + Im1 z * Im2 (z3) -
  Im2 z * Im1 (z3))*] by Th26
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * Im1 (z3)+ Im1 z * Rea (z3) + Im2 z * Im3 (
z3) - Im3 z * Im2 (z3)), (Rea z * Im2 (z3) + Im2 z * Rea (z3) + Im3 z * Im1 (z3
) - Im1 z * Im3 (z3)), (Rea z * Im3 (z3) + Im3 z * Rea (z3) + Im1 z * Im2 (z3)
  - Im2 z * Im1 (z3))*] by Th27
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z)+ Im1 z * Rea (1/zz * z*') +
Im2 z * Im3 (1/zz * z*') - Im3 z * Im2 (1/zz * z*')), (Rea z * Im2 (1/zz * z*')
+ Im2 z * Rea (1/zz * z*') + Im3 z * Im1 (1/zz * z*') - Im1 z * Im3 (1/zz * z*'
)), (Rea z * Im3 (1/zz * z*') + Im3 z * Rea (1/zz * z*') + Im1 z * Im2 (1/zz *
  z*') - Im2 z * Im1 (1/zz * z*'))*] by Th25
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z)+ Im1 z * (1/zz*Rea z) + Im2
  z * Im3 (1/zz * z*') - Im3 z * Im2 (1/zz * z*')), (Rea z * Im2 (1/zz * z*') +
Im2 z * Rea (1/zz * z*') + Im3 z * Im1 (1/zz * z*') - Im1 z * Im3 (1/zz * z*'))
, (Rea z * Im3 (1/zz * z*') + Im3 z * Rea (1/zz * z*') + Im1 z * Im2 (1/zz * z
  *') - Im2 z * Im1 (1/zz * z*'))*] by Th24
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z)+ Im1 z * (1/zz*Rea z) + Im2
  z * (-1/zz*Im3 z) - Im3 z * Im2 (z3)), (Rea z * Im2 (z3) + Im2 z * Rea (z3) +
  Im3 z * Im1 (z3) - Im1 z * Im3 (z3)), (Rea z * Im3 (z3) + Im3 z * Rea (z3) +
  Im1 z * Im2 (z3) - Im2 z * Im1 (z3))*] by Th27
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z)+ Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * Im2 (z3) + Im2 z * Rea (z3
) + Im3 z * Im1 (z3) - Im1 z * Im3 (z3)), (Rea z * Im3 (z3) + Im3 z * Rea (z3)
  + Im1 z * Im2 (z3) - Im2 z * Im1 (z3))*] by Th26
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z)+ Im1 z * (1/zz*Rea z) + Im2
  z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z *
Rea (z3) + Im3 z * Im1 (z3) - Im1 z * Im3 (z3)),(Rea z * Im3 (z3) + Im3 z * Rea
  (z3) + Im1 z * Im2 (z3) - Im2 z * Im1 (z3))*] by Th26
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z)+ Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z * (1
/zz*Rea z) + Im3 z * Im1 (z3) - Im1 z * Im3 (z3)), (Rea z * Im3 (z3) + Im3 z *
  Rea (z3) + Im1 z * Im2 (z3) - Im2 z * Im1 (1/zz * z*'))*] by Th24
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z) + Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z * (1
/zz*Rea z) + Im3 z * (-1/zz*Im1 z) - Im1 z * Im3 (z3)), (Rea z * Im3 (z3) + Im3
  z * Rea (z3) + Im1 z * Im2 (z3) - Im2 z * Im1 (z3))*] by Th25
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z) + Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z * (1
/zz*Rea z) + Im3 z * (-1/zz*Im1 z) - Im1 z * (-1/zz*Im3 z)), (Rea z * Im3 (1/zz
  * z*') + Im3 z * Rea (z3) + Im1 z * Im2 (z3) - Im2 z * Im1 (z3))*] by Th27
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z) + Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z * (1
  /zz*Rea z) + Im3 z * (-1/zz*Im1 z) - Im1 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*
Im3 z) + Im3 z * Rea (1/zz * z*') + Im1 z * Im2 (1/zz * z*') - Im2 z * Im1 (1/
  zz * z*'))*] by Th27
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z) + Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z * (1
  /zz*Rea z) + Im3 z * (-1/zz*Im1 z) - Im1 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*
Im3 z) + Im3 z * (1/zz*Rea z) + Im1 z * Im2 (1/zz * z*') - Im2 z * Im1 (1/zz *
  z*'))*] by Th24
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z) + Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z * (1
  /zz*Rea z) + Im3 z * (-1/zz*Im1 z) - Im1 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*
Im3 z) + Im3 z * (1/zz*Rea z) + Im1 z * (-1/zz*Im2 z) - Im2 z * Im1 (1/zz * z*'
  ))*] by Th26
    .=[*(Rea z * (1/zz*Rea z) - Im1 z * (-1/zz*Im1 z) - Im2 z * (-1/zz*Im2 z
) - Im3 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*Im1 z) + Im1 z * (1/zz*Rea z) + Im2
z * (-1/zz*Im3 z) - Im3 z * (-1/zz*Im2 z)), (Rea z * (-1/zz*Im2 z) + Im2 z * (1
  /zz*Rea z) + Im3 z * (-1/zz*Im1 z) - Im1 z * (-1/zz*Im3 z)), (Rea z * (-1/zz*
Im3 z) + Im3 z * (1/zz*Rea z) + Im1 z * (-1/zz*Im2 z) - Im2 z * (-1/zz*Im1 z))
  *] by Th25
    .=[*((Rea z)^2 + (Im1 z)^2 + (Im2 z)^2 + (Im3 z)^2)/zz, 0,0,0*]
    .=[*zz/zz, 0,0,0*] by Th11;
  hence thesis;
end;
