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

theorem
  z"=[*Rea z/|.z.|^2,-Im1 z/|.z.|^2, -Im2 z/|.z.|^2,-Im3 z/|.z.|^2*]
proof
 z"=[*Rea z",Im1 z",Im2 z",Im3 z"*] by QUATERNI:24
    .=[*(Rea z)/(|.z.|^2),Im1 z",Im2 z",Im3 z"*] by QUATERN2:29
    .=[*(Rea z)/(|.z.|^2),-(Im1 z)/(|.z.|^2),Im2 z",Im3 z"*] by QUATERN2:29
    .=[*(Rea z)/(|.z.|^2),-(Im1 z)/(|.z.|^2),-(Im2 z)/(|.z.|^2),Im3 z"*] by
QUATERN2:29
    .=[*(Rea z)/(|.z.|^2),-(Im1 z)/(|.z.|^2), -(Im2 z)/(|.z.|^2),-(Im3 z)/(
  |.z.|^2)*] by QUATERN2:29;
  hence thesis;
end;
