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

theorem
  (z1-z2)*(z3-z4)=(z2-z1)*(z4-z3)
proof
 (z2-z1)*(z4-z3)= (z2-z1)*z4-(z2-z1)*z3 by QUATERN2:24
    .=z2*z4-z1*z4-(z2-z1)*z3 by QUATERN2:23
    .=z2*z4-z1*z4-(z2*z3-z1*z3) by QUATERN2:23
    .=z2*z4-z1*z4-z2*z3+z1*z3 by Th55
    .=z1*z3-z2*z3-z1*z4+z2*z4 by Th70;
  hence thesis by Th57;
end;
