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

theorem
  z^3=-(-z)^3
proof
A1: z^3=z*z^2 by QUATERN2:16;
  (-z)^3 =(-z)*(-z)^2 by QUATERN2:16
    .=(-z)*z^2 by Th81
    .=((-1q) *z)* z^2 by QUATERN2:19
    .=(-1q)*(z*z^2) by QUATERN2:16
    .=-z^3 by A1,QUATERN2:19;
  hence thesis;
end;
