
theorem pr0:
for F being Field,
    E being FieldExtension of F
for p being Element of the carrier of Polynom-Ring F
for q being Element of the carrier of Polynom-Ring E st p = q
holds NormPolynomial p = NormPolynomial q
proof
let F be Field, E be FieldExtension of F;
let p be Element of the carrier of Polynom-Ring F;
let q be Element of the carrier of Polynom-Ring E;
assume AS: p = q;
set np = NormPolynomial p, nq = NormPolynomial q;
B1: F is Subfield of E by FIELD_4:7;
B: F is Subring of E by FIELD_4:def 1;
per cases;
suppose q is zero;
   then B: q = 0_.(E) by UPROOTS:def 5;
   then A: nq = 0_.(E) by RING_4:22 .= 0_.(F) by FIELD_4:12;
   p = 0_.(F) by AS,B,FIELD_4:12;
   hence thesis by A,RING_4:22;
   end;
suppose q is non zero; then
LC q is non zero; then
A1: q.(len q-'1) is non zero by RATFUNC1:def 6;
len p - 1 = deg p by HURWITZ:def 2
         .= deg q by AS,FIELD_4:20
         .= len q - 1 by HURWITZ:def 2; then
A: (p.(len p-'1))" = (q.(len q-'1))" by AS,A1,B1,Th19f;
now let n be Element of NAT;
  thus np.n = p.n / p.(len p-'1) by POLYNOM5:def 11
           .= q.n / q.(len q-'1) by B,A,AS,Th18
           .= nq.n by POLYNOM5:def 11;
  end;
hence thesis;
end;
end;
