
theorem Th42:
  for f being Polynomial of F_Complex holds deg(f*') = deg(f)
proof
  let f be Polynomial of F_Complex;
A1: for k being Nat holds f.k = 0.F_Complex iff (f*').k = 0.F_Complex
  proof
    let k be Nat;
    reconsider k1 = k as Element of NAT by ORDINAL1:def 12;
    hereby
      assume f.k = 0.F_Complex;
      hence (f*').k = power(F_Complex).(-1_F_Complex,k1) * 0.F_Complex by Def9,
COMPLFLD:47
        .= 0.F_Complex;
    end;
    assume (f*').k = 0.F_Complex;
    then
A2: 0.F_Complex = power(F_Complex).(-1_F_Complex,k1) * ((f.k)*') by Def9;
    power(F_Complex).(-1_F_Complex,k1) <> 0.F_Complex by Th2;
    then (f.k)*' = 0.F_Complex by A2,VECTSP_1:12;
    hence thesis by COMPLEX1:28,COMPLFLD:7;
  end;
A3: now
    assume
A4: len f > len(f*');
    then len f + 1 > 0 + 1 by XREAL_1:6;
    then len f >= 1 by NAT_1:13;
    then reconsider l = len(f)-1 as Element of NAT by INT_1:5;
    l + 1 > len(f*') by A4;
    then l >= len(f*') by NAT_1:13;
    then
A5: (f*').l = 0.F_Complex by ALGSEQ_1:8;
    len f = l + 1;
    then f.l <> 0.F_Complex by ALGSEQ_1:10;
    hence contradiction by A1,A5;
  end;
  now
    let i be Nat;
    assume i >= len f;
    then f.i = 0.F_Complex by ALGSEQ_1:8;
    hence (f*').i = 0.F_Complex by A1;
  end;
  then len f is_at_least_length_of (f*') by ALGSEQ_1:def 2;
  then len f >= len (f*') by ALGSEQ_1:def 3;
  hence thesis by A3,XXREAL_0:1;
end;
