reserve n for Nat;

theorem lemacf2:
for R being domRing,
    B1,B2 being non zero bag of the carrier of R
for p being (Ppoly of R,B1),
    q being Ppoly of R,B2 holds p *' q is Ppoly of R,(B1+B2)
proof
let R be domRing, B1,B2 be non zero bag of the carrier of R;
set B = B1 + B2;
let p be (Ppoly of R,B1), q be Ppoly of R,B2;
reconsider r = p *' q as Ppoly of R by lemppoly3;
p <> 0_.(R) & q <> 0_.(R); then
A: deg r = deg p + deg q by HURWITZ:23
        .= card BRoots p + deg q by lemacf5
        .= card B1 + deg q by pf2
        .= card B1 + card BRoots q by lemacf5
        .= card B1 + card B2 by pf2
        .= card B by UPROOTS:15;
now let c be Element of R;
  thus multiplicity(r,c)
           = multiplicity(p,c) + multiplicity(q,c) by UPROOTS:55
          .= (BRoots p).c + multiplicity(q,c) by UPROOTS:def 9
          .= B1.c + multiplicity(q,c) by pf2
          .= B1.c + (BRoots q).c by UPROOTS:def 9
          .= B1.c + B2.c by pf2
          .= B.c by PRE_POLY:def 5;
  end;
hence thesis by A,dpp;
end;
