
theorem Th37:
  for L being non degenerated comRing, x being Element of L, q
being non-zero (Polynomial of L), i being Nat holds len ((<%x, 1.L%>
  `^i)*'q) = i + len q
proof
  let L being non degenerated comRing, x being Element of L, q being non-zero
  Polynomial of L;
  set r = <%x, 1.L%>;
  defpred P[Nat] means len ((r`^$1)*'q) = $1 + len q;
A1: for i being Nat st P[i] holds P[i+1]
  proof
    let i be Nat such that
A2: P[i];
    len q > 0 by Th14;
    then
A3: (r`^i)*'q is non-zero by A2,Th14;
    thus len ((r`^(i+1))*'q) = len ((r`^1)*'(r`^i)*'q) by Th27
      .= len (r*'(r`^i)*'q) by POLYNOM5:16
      .= len (r*'((r`^i)*'q)) by POLYNOM3:33
      .= (i+len q)+1 by A2,A3,Th35
      .= (i+1)+len q;
  end;
  len ((r`^0)*'q) = len ((1_. L)*'q) by POLYNOM5:15
    .= 0 qua Nat+ len q by POLYNOM3:35;
  then
A4: P[ 0 ];
  thus for i being Nat holds P[i] from NAT_1:sch 2(A4,A1);
end;
