reserve n for Nat;

theorem m0:
for R being domRing,
    S being non empty finite Subset of R
for p being Ppoly of R,S
for a being Element of R st a in S
holds rpoly(1,a) divides p & not rpoly(1,a)`^2 divides p
proof
let R be domRing, S be non empty finite Subset of R;
let p being Ppoly of R,S; let a being Element of R;
assume a in S;
then A: (Bag S).a = 1 by UPROOTS:7;
X: rpoly(1,a) `^ ((Bag S).a) divides p &
   not rpoly(1,a) `^ ((Bag S).a + 1) divides p by pf1;
hence rpoly(1,a) divides p by A,POLYNOM5:16;
thus thesis by A,X;
end;
