reserve n for Nat;

theorem BR5aa:
for R being domRing,
    a being Element of R holds multiplicity(rpoly(1,a),a) = 1
proof
let R be domRing, a be Element of R;
set p = rpoly(1,a);
p *' 1_.(R) = p;
then p divides p by RING_4:1;
then A: p`^1 divides p by POLYNOM5:16;
p <> 0_.(R);
then deg(p *' p) = deg(p) + deg(p) by HURWITZ:23
                .= deg(p) + 1 by HURWITZ:27
                .= 1 + 1 by HURWITZ:27;
then deg(p *' p) > 1;
then B: deg(p *' p) > deg(p) by HURWITZ:27;
p *' p = p`^2 by POLYNOM5:17;
then not p`^(1+1) divides p by B,prl25;
hence multiplicity(p,a) = 1 by A,multip;
end;
