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

theorem
  |.z.| ^2 = |.z*z*'.|
proof
  |.z*z.| = |.z*z*'.| & |.z.| ^2 =
    (Rea z)^2 + (Im1 z)^2 + (Im2 z)^2 + (Im3 z) ^2 by Th11,QUATERNI:89;
  hence thesis by QUATERNI:88;
end;
