reserve n for Nat;

theorem lemacf:
for R being domRing,
    a being Element of R holds rpoly(1,a) is Ppoly of R,Bag{a}
proof
let R be domRing, a be Element of R;
reconsider p = rpoly(1,a) as Ppoly of R by lemppoly1;
A: deg p = 1 by HURWITZ:27 .= card {a} by CARD_1:30
        .= card ({a},1)-bag by UPROOTS:13;
now let c be Element of R;
  per cases;
  suppose B: c = a;
    then C: c in {a} by TARSKI:def 1;
    thus multiplicity(p,c) = 1 by B,BR5aa .= (Bag{a}).c by C,UPROOTS:7;
    end;
  suppose B: c <> a;
    then C: not c in {a} by TARSKI:def 1;
    thus multiplicity(p,c) = 0 by B,BR5aaa .= (Bag{a}).c by C,UPROOTS:6;
    end;
  end;
hence thesis by A,dpp;
end;
