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 QUATERN2:17
    .=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 QUATERN2:2;
  hence thesis;
end;
