
theorem Th44:
  for f being Polynomial of F_Complex holds (-f)*' = -(f*')
proof
  let f be Polynomial of F_Complex;
  set h1 = -f;
A1: now
    let k9 be object;
    assume k9 in dom(h1*');
    then reconsider k = k9 as Element of NAT;
    h1.k = -f.k by BHSP_1:44;
    then (h1*').k = power(F_Complex).(-1_F_Complex,k) * (-f.k)*' by Def9
      .= power(F_Complex).(-1_F_Complex,k) * (-((f.k)*')) by COMPLFLD:52
      .= -(power(F_Complex).(-1_F_Complex,k) * ((f.k)*')) by VECTSP_1:8
      .= -((f*').k) by Def9;
    hence (h1*').k9 = (-(f*')).k9 by BHSP_1:44;
  end;
  dom(h1*') = NAT by FUNCT_2:def 1
    .= dom(-(f*')) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
