
theorem uu5:
for F1 being Field,
    F2 being F1-monomorphic F1-homomorphic Field
for h being Monomorphism of F1,F2
for p being Element of the carrier of Polynom-Ring F1
holds NormPolynomial((PolyHom h).p) = (PolyHom h).(NormPolynomial p)
proof
let F1 be Field, F2 be F1-monomorphic F1-homomorphic Field;
let h be Monomorphism of F1,F2;
let p be Element of the carrier of Polynom-Ring F1;
set q = NormPolynomial((PolyHom h).p), pp = (PolyHom h).p,
    r = (PolyHom h).(NormPolynomial p);
per cases;
suppose p is zero; then
  A: p = 0_.(F1) by UPROOTS:def 5; then
  NormPolynomial p = 0_.(F1) by RING_4:22; then
  B: (PolyHom h).(NormPolynomial p) = 0_.(F2) by FIELD_1:22;
  pp = 0_.(F2) by A,FIELD_1:22;
  hence thesis by B,RING_4:22;
end;
suppose p is non zero; then
LC p is non zero; then
B: p.(len p-'1) <> 0.F1 by RATFUNC1:def 6;
deg pp = deg p by FIELD_1:31; then
deg p = len pp - 1 by HURWITZ:def 2; then
A: len p - 1 = len pp - 1 by HURWITZ:def 2;
now let n be Nat;
  reconsider i = n as Element of NAT by ORDINAL1:def 12;
  pp.i = h.(p.i) by FIELD_1:def 2; then
  q.i = (h.(p.i)) / pp.(len p-'1) by A,POLYNOM5:def 11
     .= h.(p.i) * (h.(p.(len p-'1)))" by FIELD_1:def 2
     .= h.(p.i) * h.((p.(len p-'1))") by B,FIELD_2:6
     .= h.( (p.i) / p.(len p-'1) ) by GROUP_6:def 6
     .= h.( (NormPolynomial p).i ) by POLYNOM5:def 11
     .= r.i by FIELD_1:def 2;
  hence q.n = r.n;
  end;
hence thesis;
end;
end;
