reserve n for Nat;

theorem divi1b:
for R being domRing,
    p being non zero Polynomial of R
for a being Element of R,
    b being non zero Element of R
holds rpoly(1,a) divides p iff rpoly(1,a) divides (b * p)
proof
let F be domRing, p be non zero Polynomial of F;
let a be Element of F, b be non zero Element of F;
set q = rpoly(1,a);
now assume q divides (b*p);
  then 0.F = eval(b*p,a) by Th9 .= b * eval(p,a) by Th30;
  then eval(p,a) = 0.F by VECTSP_2:def 1;
  hence q divides p by Th9;
  end;
hence thesis by divi1;
end;
