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

theorem
  (z1+z2)*(z3-z4)=z1*z3+z2*z3-z1*z4-z2*z4
proof
 (z1+z2)*(z3-z4)= (z1+z2)*z3 -(z1+z2)*z4 by QUATERN2:24
    .=z1*z3+z2*z3-(z1+z2)*z4 by QUATERN2:18
    .=z1*z3+z2*z3-(z1*z4+z2*z4) by QUATERN2:18
    .=z1*z3+z2*z3-z1*z4-z2*z4 by Th54;
  hence thesis;
end;
