reserve n for Nat;

theorem lemppoly3:
for R being domRing,
    p,q being Ppoly of R holds p *' q is Ppoly of R
proof
let R be domRing, p,q be Ppoly of R;
consider Fp being non empty FinSequence of Polynom-Ring R such that
B0: p = Product Fp & for i being Nat st i in dom Fp
                     ex a being Element of R st Fp.i = rpoly(1,a) by dpp1;
consider Fq being non empty FinSequence of Polynom-Ring R such that
B1: q = Product Fq & for i being Nat st i in dom Fq
                     ex a being Element of R st Fq.i = rpoly(1,a) by dpp1;
set G = Fp ^ Fq;
A: now let i be Nat;
    assume AS: i in dom G;
    per cases by AS,FINSEQ_1:25;
    suppose B2: i in dom Fp;
      then Fp.i = G.i by FINSEQ_1:def 7;
      hence ex a being Element of R st G.i = rpoly(1,a) by B0,B2;
      end;
    suppose ex n being Nat st n in dom Fq & i =len Fp + n;
      then consider n being Nat such that
      B2: n in dom Fq & i = len Fp + n;
      G.i = Fq.n by B2,FINSEQ_1:def 7;
      hence ex a being Element of R st G.i = rpoly(1,a) by B1,B2;
      end;
    end;
Product G = (Product Fp) * (Product Fq) by GROUP_4:5
         .= p *' q by B0,B1,POLYNOM3:def 10;
hence thesis by A,dpp1;
end;
