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

theorem Th11:
  |.z.| ^2 = (Rea z)^2 + (Im1 z)^2 + (Im2 z)^2 + (Im3 z)^2
proof
  (Rea z)^2 + (Im1 z)^2 + (Im2 z)^2 + (Im3 z)^2 >= 0 by QUATERNI:74; then
  |.z.|^2= sqrt(((Rea z)^2 + (Im1 z)^2 + (Im2 z)^2 + (Im3 z)^2)^2) by
SQUARE_1:29
    .= (Rea z)^2 + (Im1 z)^2 + (Im2 z)^2 + (Im3 z)^2
      by QUATERNI:74,SQUARE_1:22;
  hence thesis;
end;
