theorem Th53:
  for a, b being Complex holds (a-b)*z = a*z - b*z
proof
  let a, b be Complex;
  reconsider aa = a, bb = b as Element of COMPLEX by XCMPLX_0:def 2;
  (a - b)*z=(a+(-b))*z .= aa*z +(-bb)*z by Th52
    .= a*z - b*z by Th45;
  hence thesis;
end;
