 reserve n for Nat;

theorem Th9:
  for R being non degenerated Ring,
      n being non zero Element of NAT holds anpoly(1.R,n) = rpoly(n,0.R)
proof
  let R be non degenerated Ring, n be non zero Element of NAT;
  set p = anpoly(1.R,n), r = rpoly(n,0.R);
  now let i be Element of NAT;
A1: 1 <= n by NAT_1:53;
  per cases;
  suppose A2: i = 0;
    then r.i = -power(R).(0.R,n) by HURWITZ:25
            .= -((0.R)|^n) by BINOM:def 2
            .= -0.R by A1,EC_PF_2:5
            .= p.i by A2,POLYDIFF:25;
    hence p.i = r.i;
    end;
  suppose A3: i = n;
    then r.i = 1_R by HURWITZ:25
            .= p.i by A3,POLYDIFF:24;
    hence p.i = r.i;
    end;
  suppose A4: i <> 0 & i <> n;
    then r.i = 0.R by HURWITZ:26
            .= p.i by A4,POLYDIFF:25;
    hence p.i = r.i;
    end;
  end;
  hence thesis;
end;
