
theorem field426a:
for n being Nat
for F being Field, E being FieldExtension of F,
    p being Polynomial of n,F, q being Polynomial of n,E
st p = q holds Support q = Support p
proof
let n be Nat, F be Field, E be FieldExtension of F,
    p be Polynomial of n,F, q be Polynomial of n,E;
assume AS: p = q;
H1: F is Subring of E by FIELD_4:def 1;
A: now let o be object;
   assume A1: o in Support p; then
   reconsider b = o as Element of Bags n;
   p.b <> 0.F by A1,POLYNOM1:def 4; then
   p.b <> 0.E by H1,C0SP1:def 3;
   hence o in Support q by AS,POLYNOM1:def 4;
   end;
now let o be object;
   assume A1: o in Support q; then
   reconsider b = o as Element of Bags n;
   q.b <> 0.E by A1,POLYNOM1:def 4; then
   q.b <> 0.F by H1,C0SP1:def 3;
   hence o in Support p by AS,POLYNOM1:def 4;
   end;
hence thesis by A,TARSKI:2;
end;
