theorem Th43:
  for a being Real, z being Complex holds Im(a*z) = a*Im(z)
proof
  let a be Real;
  let z be Complex;
  Im(a * z) = Re a * Im z + Re z * Im a by COMPLEX1:9
    .= Re a * Im z + Re z * 0 by COMPLEX1:def 2
    .= a * Im (z) by COMPLEX1:def 1;
  hence thesis;
end;
