
theorem
for R being domRing,
    S being domRingExtension of R
for p being non zero Element of the carrier of Polynom-Ring R
for h being R-fixing Monomorphism of S
holds h|Roots(S,p) is Permutation of Roots(S,p)
proof
let R be domRing, S be domRingExtension of R;
let p be non zero Element of the carrier of Polynom-Ring R;
let h be R-fixing Monomorphism of S;
set N = Roots(S,p);
reconsider g = h|Roots(S,p) as Function of N,S by FUNCT_2:32;
C: dom g = Roots(S,p) by FUNCT_2:def 1;
now let o be object;
   assume o in rng g; then
   consider b being object such that 
   B1: b in dom g & o = g.b by FUNCT_1:def 3;
   reconsider b as Element of S by C,B1;
   h.b in Roots(S,p) by B1,prm;
   hence o in Roots(S,p) by B1,FUNCT_1:47;
   end; 
then rng g c= Roots(S,p);
then reconsider g as Function of N,N by C,FUNCT_2:2;
E: card Roots(S,p) = card Roots(S,p);
A: g is one-to-one by FUNCT_1:52; then
g is onto by E,FINSEQ_4:63;
hence thesis by A;
end;
