 reserve n for Nat;

theorem Th13:
  for R being non degenerated comRing
   holds (<%0.R,1.R%>)`^n = anpoly(1.R,n)
   proof
     let R be non degenerated comRing;
     defpred P[Nat] means (<%0.R,1.R%>)`^$1= anpoly(1.R,$1);
A1:  P[0]
     proof
       thus (<%0.R,1.R%>)`^0 = 1_.R by POLYNOM5:15
                    .= anpoly(1.R,0);
     end;
A2:  now let n be Nat;
      assume P[n]; then
      (<%0.R,1.R%>)`^(n+1) = anpoly(1.R,n) *' <%-0.R,1.R%> by POLYNOM5:19
                        .= anpoly(1.R,n) *' rpoly(1,0.R) by RING_5:10
                        .= anpoly(1.R,n) *' anpoly(1.R,1) by Th9
                        .= anpoly(1.R*1.R,n+1) by Th11
                        .= anpoly(1.R,n+1);
      hence P[n+1];
    end;
    for k being Nat holds P[k] from NAT_1:sch 2(A1,A2);
    hence thesis;
   end;
