
theorem Th43:
  for f being Polynomial of F_Complex for z being Element of
  F_Complex holds (z * f)*' = (z*') * (f*')
proof
  let f be Polynomial of F_Complex;
  let x be Element of F_Complex;
  set g1 = x * f, g2 = (x*') * (f*');
A1: now
    let k9 be object;
    assume k9 in dom(g1*');
    then reconsider k = k9 as Element of NAT;
    g1.k = x * f.k by POLYNOM5:def 4;
    then (g1*').k = power(F_Complex).(-1_F_Complex,k) * (x * f.k)*' by Def9
      .= power(F_Complex).(-1_F_Complex,k) * ((x*') * ((f.k)*')) by COMPLFLD:54
      .= (x*') * (power(F_Complex).(-1_F_Complex,k) * ((f.k)*'))
      .= (x*') * (f*').k by Def9;
    hence (g1*').k9 = g2.k9 by POLYNOM5:def 4;
  end;
  dom(g1*') = NAT by FUNCT_2:def 1
    .= dom g2 by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
