
theorem Th36:
  for L being non degenerated comRing, x being Element of L, i
  being Nat holds len (<%x, 1.L%>`^i) = i+1
proof
  let L be non degenerated comRing, x be Element of L;
  defpred P[Nat] means len (<%x, 1.L%>`^$1) = $1+1;
  set r = <%x, 1.L%>;
A1: for i being Nat st P[i] holds P[i+1]
  proof
    let i be Nat such that
A2: P[i];
    reconsider ri = r`^i as non-zero Polynomial of L by A2,Th14;
    thus len (r`^(i+1)) = len ((r`^1)*'ri) by Th27
      .= len (r*'ri) by POLYNOM5:16
      .= i+1+1 by A2,Th35;
  end;
  r`^0 = 1_. L by POLYNOM5:15;
  then
A3: P[ 0 ] by POLYNOM4:4;
  thus for i being Nat holds P[i] from NAT_1:sch 2(A3,A1);
end;
