
theorem Th57:
  for L be Field for p be Polynomial of L st len p <> 0 holds len
  NormPolynomial(p) = len p
proof
  let L be Field;
  let p be Polynomial of L;
  assume len p <> 0;
  then len p >= 0+1 by NAT_1:13;
  then len p = len p-'1+1 by XREAL_1:235;
  then p.(len p-'1) <> 0.L by ALGSEQ_1:10;
  then
A1: (p.(len p-'1))" <> 0.L by VECTSP_1:25;
A2: now
    let n be Nat;
    assume
A3: n is_at_least_length_of NormPolynomial(p);
    n is_at_least_length_of p
    proof
      let i be Nat;
      reconsider ii = i as Element of NAT by ORDINAL1:def 12;
      assume i >= n;
      then (NormPolynomial(p)).i = 0.L by A3;
      then p.ii / p.(len p-'1) = 0.L by Def11;
      then p.ii * (p.(len p-'1))" = 0.L;
      hence thesis by A1,VECTSP_1:12;
    end;
    hence len p <= n by ALGSEQ_1:def 3;
  end;
  len p is_at_least_length_of NormPolynomial(p)
  proof
    let n be Nat;
    assume
A4: n >= len p;
    reconsider nn = n as Element of NAT by ORDINAL1:def 12;
    thus (NormPolynomial(p)).n = p.nn / p.(len p-'1) by Def11
      .= 0.L / p.(len p-'1) by A4,ALGSEQ_1:8
      .= 0.L * (p.(len p-'1))"
      .= 0.L;
  end;
  hence thesis by A2,ALGSEQ_1:def 3;
end;
