
theorem co:
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 q = p holds Coeff q = Coeff p
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: q = p;
H: F is Subfield of E by FIELD_4:7;
A: now let o be object;
   assume o in Coeff q; then
   o in {q.i where i is Element of NAT : q.i <> 0.E} by FIELD_7:def 3;
   then consider i being Element of NAT such that
   A1: o = q.i & q.i <> 0.E;
   o = p.i & p.i <> 0.F by A1,AS,H,EC_PF_1:def 1; then
   o in {p.i where i is Element of NAT : p.i <> 0.F};
   hence o in Coeff p by FIELD_7:def 3;
   end;
now let o be object;
   assume o in Coeff p; then
   o in {p.i where i is Element of NAT : p.i <> 0.F} by FIELD_7:def 3;
   then consider i being Element of NAT such that
   A1: o = p.i & p.i <> 0.F;
   o = q.i & q.i <> 0.E by A1,AS,H,EC_PF_1:def 1; then
   o in {q.i where i is Element of NAT : q.i <> 0.E};
   hence o in Coeff q by FIELD_7:def 3;
   end;
hence thesis by A,TARSKI:2;
end;
