
theorem Th58:
  for L be Field for p be Polynomial of L st len p <> 0 for x be
  Element of L holds eval(NormPolynomial(p),x) = eval(p,x)/p.(len p-'1)
proof
  let L be Field;
  let p be Polynomial of L;
  assume
A1: len p <> 0;
  set NPp = NormPolynomial(p);
  let x be Element of L;
  consider F1 be FinSequence of the carrier of L such that
A2: eval(p,x) = Sum F1 and
A3: len F1 = len p and
A4: for n be Element of NAT st n in dom F1 holds F1.n = p.(n-'1) * (
  power L).(x,n-'1) by POLYNOM4:def 2;
  consider F2 be FinSequence of the carrier of L such that
A5: eval(NPp,x) = Sum F2 and
A6: len F2 = len NPp and
A7: for n be Element of NAT st n in dom F2 holds F2.n = NPp.(n-'1) * (
  power L).(x,n-'1) by POLYNOM4:def 2;
  len F1 = len F2 by A1,A3,A6,Th57;
  then
A8: dom F1 = dom F2 by FINSEQ_3:29;
  now
    let i be object;
    assume
A9: i in dom F1;
    then reconsider i1=i as Element of NAT;
A10: p.(i1-'1) * (power L).(x,i1-'1) = F1.i by A4,A9
      .= F1/.i by A9,PARTFUN1:def 6;
    thus F2/.i = F2.i by A8,A9,PARTFUN1:def 6
      .= NPp.(i1-'1) * (power L).(x,i1-'1) by A7,A8,A9
      .= p.(i1-'1) / p.(len p-'1) * (power L).(x,i1-'1) by Def11
      .= p.(i1-'1) * (p.(len p-'1))" * (power L).(x,i1-'1)
      .= (F1/.i)*(p.(len p-'1))" by A10,GROUP_1:def 3;
  end;
  then F2 = F1*(p.(len p-'1))" by A8,POLYNOM1:def 2;
  then eval(NormPolynomial(p),x) = eval(p,x) * (p.(len p-'1))" by A2,A5,
POLYNOM1:13;
  hence thesis;
end;
