reserve n for Nat;

theorem multipp0:
for R being non degenerated comRing,
    p being non zero Polynomial of R
for a being Element of R
holds multiplicity(p,a) = 0 iff not rpoly(1,a) divides p
proof
let R be non degenerated comRing, p be non zero Polynomial of R;
let a be Element of R;
A: now assume multiplicity(p,a) = 0;
   then C: not(a is_a_root_of p) by UPROOTS:52;
   now assume rpoly(1,a) divides p;
     then consider s being Polynomial of R such that
     B:  p = rpoly(1,a) *' s by RING_4:1;
     thus contradiction by C,B,prl2,HURWITZ:30;
     end;
   hence not rpoly(1,a) divides p;
   end;
now assume multiplicity(p,a) <> 0;
  then multiplicity(p,a) + 1 > 0 + 1 by XREAL_1:6;
  then multiplicity(p,a) >= 1 by NAT_1:13; then
  ex s being Polynomial of R st p = rpoly(1,a) *' s by UPROOTS:52,HURWITZ:33;
  hence rpoly(1,a) divides p by RING_4:1;
  end;
hence thesis by A;
end;
