reserve n for Nat;

theorem
for R being domRing,
    S being non empty finite Subset of R,
    p being Ppoly of R,S
holds Roots(p) = S
proof
let R be domRing, S be non empty finite Subset of R,
    p be Ppoly of R,S;
A0: now let o be object;
    assume AS: o in S;
    then reconsider x = o as Element of R;
    eval(p,x) = 0.R by AS,m1;
    then x is_a_root_of p by POLYNOM5:def 7;
    hence o in Roots(p) by POLYNOM5:def 10;
    end;
then card(Roots(p) \ S) = card Roots(p) - card S by TARSKI:def 3,CARD_2:44;
then B: (card Roots(p) - card S) + card S >= 0 + card S by XREAL_1:6;
card Roots(p) <= deg p by degpoly;
then card Roots(p) <= card S by m00;
then card S = card Roots(p) by B,XXREAL_0:1;
hence thesis by A0,CARD_2:102,TARSKI:def 3;
end;
