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

theorem
  (1/(|.z.| ^2) * z*')=[*1/(|.z.|^2)*Rea z, -1/(|.z.|^2)*Im1 z, -1/(|.z
  .|^2)*Im2 z, -1/(|.z.|^2)*Im3 z*]
proof
  set zz = |.z.| ^2;
 (1/zz * z*')=[*Rea ((1/zz * z*')),Im1 ((1/zz * z*')),Im2 ((1/zz * z*')),
  Im3 ((1/zz * z*'))*] by QUATERNI:24; then
 (1/zz * z*')=[*1/zz*Rea z, Im1 ((1/zz * z*')),Im2 ((1/zz * z*')),Im3 ((1
  /zz * z*'))*] by Th24; then
  (1/zz * z*')=[*1/zz*Rea z, -1/zz*Im1 z, Im2 ((1/zz * z*')),Im3 ((1/zz *
  z*'))*] by Th25; then
 (1/zz * z*')=[*1/zz*Rea z, -1/zz*Im1 z, -1/zz*Im2 z, Im3 ((1/zz * z*'))
  *] by Th26;
  hence thesis by Th27;
end;
