theorem Th42:
  for a being Real, z being Complex holds Re(a*z)=a* Re(z)
proof
  let a be Real;
  let z be Complex;
  Re(a * z) = Re a * Re z - Im a * Im z by COMPLEX1:9
    .= Re a * Re z - 0 * Im z by COMPLEX1:def 2
    .= a * Re (z) by COMPLEX1:def 1;
  hence thesis;
end;
