reserve n for Nat;

theorem multip1d:
for R being domRing,
    p being non zero Polynomial of R
for b being non zero Element of R,
    a being Element of R holds multiplicity(p,a) = multiplicity(b*p,a)
proof
let F be domRing, p be non zero Polynomial of F;
let b be non zero Element of F, a be Element of F;
set r = rpoly(1,a), np = multiplicity(p,a);
r`^np divides (b*p) & not r`^(np+1) divides p by multip,divi1;
then r`^np divides (b*p) & not r`^(np+1) divides (b*p) by divi1ad;
hence thesis by multip;
end;
