reserve n for Nat;

theorem
for R being domRing,
    B being non zero bag of the carrier of R,
    p being Ppoly of R,B
for a being Element of R st a in support B holds eval(p,a) = 0.R
proof
let R be domRing, F be non zero bag of the carrier of R,
    p be Ppoly of R,F; let a being Element of R;
assume a in support F;
then F.a <> 0 by PRE_POLY:def 7;
then F.a + 1 > 0 + 1 by XREAL_1:6;
then F.a >= 1 by NAT_1:13;
then multiplicity(p,a) >= 1 by dpp;
then consider s being Polynomial of R such that
A: p = rpoly(1,a) *' s by HURWITZ:33,UPROOTS:52;
thus thesis by A,RING_4:1,Th9;
end;
