
theorem cof1:
for F being Field
for E being FieldExtension of F
for p being Polynomial of E
st Coeff p c= the carrier of F holds p is Polynomial of F
proof
let F be Field, E be FieldExtension of F, p be Polynomial of E;
assume AS: Coeff p c= the carrier of F;
F is Subring of E by FIELD_4:def 1; then
A: 0.E = 0.F by C0SP1:def 3;
now let y be object;
  assume y in rng p;
  then consider x being object such that
  A1: x in dom p & y = p.x by FUNCT_1:def 3;
  reconsider x as Element of NAT by A1;
  per cases;
  suppose p.x = 0.F;
    hence y in the carrier of F by A1;
    end;
  suppose p.x <> 0.F;
    then y in Coeff p by A,A1;
    hence y in the carrier of F by AS;
    end;
  end; then
rng p c= the carrier of F; then
reconsider q = p as sequence of the carrier of F by FUNCT_2:6;
consider n being Nat such that
C: for i being Nat st i >= n holds p.i = 0.E by ALGSEQ_1:def 1;
for i being Nat st i >= n holds q.i = 0.F by C,A;
hence thesis by ALGSEQ_1:def 1;
end;
