reserve a,b,i,j,k,l,m,n for Nat;

theorem
  for a be non zero Real, a1 be Complex st a*a1 is Real holds a1 is Real
  proof
    let a be non zero Real, a1 be Complex;
    Im(a1*a) = Re a1 * Im a + Re a * Im a1 by COMPLEX1:9
    .= Re a1 * 0 + a * Im a1;
    hence thesis;
  end;
