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

theorem Th78:
  z^2 = [*(Rea z)^2-(Im1 z)^2-(Im2 z)^2-(Im3 z)^2, 2*(Rea z * Im1 z)
  , 2*(Rea z * Im2 z), 2*(Rea z * Im3 z)*]
proof
 z^2=[*Rea z^2,Im1 z^2,Im2 z^2,Im3 z^2*] by QUATERNI:24
    .=[*(Rea z)^2-(Im1 z)^2-(Im2 z)^2-(Im3 z)^2,Im1 z^2, Im2 z^2,Im3 z^2*]
  by QUATERNI:40
    .=[*(Rea z)^2-(Im1 z)^2-(Im2 z)^2-(Im3 z)^2, 2*(Rea z * Im1 z), Im2 z^2,
  Im3 z^2*] by QUATERNI:40
    .=[*(Rea z)^2-(Im1 z)^2-(Im2 z)^2-(Im3 z)^2, 2*(Rea z * Im1 z), 2*( Rea
  z * Im2 z),Im3 z^2*] by QUATERNI:40
    .=[*(Rea z)^2-(Im1 z)^2-(Im2 z)^2-(Im3 z)^2, 2*(Rea z * Im1 z), 2*(Rea z
  * Im2 z), 2*(Rea z * Im3 z)*] by QUATERNI:40;
  hence thesis;
end;
