reserve n for Nat;

theorem m1:
for R being domRing,
    S being non empty finite Subset of R,
    p being Ppoly of R,S
for a being Element of R st a in S holds eval(p,a) = 0.R
proof
let R be domRing, S be non empty finite Subset of R,
    p be Ppoly of R,S; let a be Element of R;
assume a in S;
then consider q being Polynomial of R such that
H: rpoly(1,a) *' q = p by m0,RING_4:1;
a is_a_root_of p by H,prl2,HURWITZ:30;
hence eval(p,a) = 0.R by POLYNOM5:def 7;
end;
