reserve n for Nat;

theorem lemppoly1:
for R being domRing, a being Element of R holds rpoly(1,a) is Ppoly of R
proof
let R being domRing, a be Element of R;
reconsider p = rpoly(1,a) as
                   Element of the carrier of Polynom-Ring R by POLYNOM3:def 10;
set F = <*p*>;
A: now let i be Nat; assume AS: i in dom F;
   dom F = { 1 } by FINSEQ_1:2,FINSEQ_1:38;
   then i = 1 by AS,TARSKI:def 1;
   hence ex a being Element of R st F.i = rpoly(1,a);
   end;
Product F = p by GROUP_4:9;
hence thesis by A,dpp1;
end;
