
theorem fixrr:
for F being Field,
    E being FieldExtension of F
for h being F-fixing Monomorphism of E
for p being non zero Element of the carrier of Polynom-Ring F
holds h .: Roots(E,p) = Roots(E,p)
proof
let F be Field, E being FieldExtension of F,
    h be F-fixing Monomorphism of E;
let p be non zero Element of the carrier of Polynom-Ring F;
reconsider g = h|Roots(E,p) as bijective Function of Roots(E,p),Roots(E,p)
   by FIELD_8:39;
A: g is onto;
Y: now let o be object;
   assume o in Roots(E,p); then
   consider b being object such that
   C: b in dom(h|Roots(E,p)) & (h|Roots(E,p)).b = o by A,FUNCT_1:def 3;
   D: b in dom h by C,RELAT_1:57;
   h.b = o by C,FUNCT_1:49;
   hence o in h.:Roots(E,p) by C,D,FUNCT_1:def 6;
   end;
now let o be object;
   assume o in h.:Roots(E,p); then
   consider x being object such that
   B: x in dom h & x in Roots(E,p) & o = h.x by FUNCT_1:def 6;
   C: x in dom(h|Roots(E,p)) by B,RELAT_1:57;
   o = (h|Roots(E,p)).x by B,FUNCT_1:49;
   hence o in Roots(E,p) by A,C,FUNCT_1:def 3;
   end;
hence thesis by Y,TARSKI:2;
end;
