
theorem ID2:
for R being domRing
for S being domRingExtension of R
for p being non zero Element of the carrier of Polynom-Ring R
holds card Roots(S,p) <= deg p
proof
let R be domRing, S be domRingExtension of R;
let p being non zero Element of the carrier of Polynom-Ring R;
the carrier of Polynom-Ring R c= the carrier of Polynom-Ring S
   by FIELD_4:10; then
reconsider q = p as Element of the carrier of Polynom-Ring S;
p <> 0_.(R); then
q <> 0_.(S) by FIELD_4:12; then
reconsider q as non zero Element of the carrier of Polynom-Ring S
   by UPROOTS:def 5;
Roots(S,p) = Roots(q) & deg p = deg q by FIELD_4:20,FIELD_7:13;
hence thesis by RING_5:22;
end;
