
theorem Th45:
  for f,g being Polynomial of F_Complex holds (f + g)*' = (f*') + (g*')
proof
  let f,g be Polynomial of F_Complex;
  set h1 = f+g;
A1: now
    let k9 be object;
    assume k9 in dom(h1*');
    then reconsider k = k9 as Element of NAT;
    h1.k = f.k + g.k by NORMSP_1:def 2;
    then (h1*').k = power(F_Complex).(-1_F_Complex,k) * (f.k + g.k)*' by Def9
      .= power(F_Complex).(-1_F_Complex,k) * ((f.k)*' + (g.k)*') by COMPLFLD:51
      .= (power(F_Complex).(-1_F_Complex,k) * (f.k)*') + (power(F_Complex).(
    -1_F_Complex,k) * (g.k)*')
      .= (f*').k + (power(F_Complex).(-1_F_Complex,k) * (g.k)*') by Def9
      .= (f*').k + (g*').k by Def9;
    hence (h1*').k9 = ((f*') + (g*')).k9 by NORMSP_1:def 2;
  end;
  dom(h1*') = NAT by FUNCT_2:def 1
    .= dom((f*') + (g*')) by FUNCT_2:def 1;
  hence thesis by A1,FUNCT_1:2;
end;
