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

theorem Th81:
  z^2 = (-z)^2
proof
 (-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))*] by Th78
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(-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:41
    .=[*(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)*];
  hence thesis by Th78;
end;
