
theorem Th55:
  for L being domRing, x be Element of L, n being Nat
  holds degree BRoots (<%-x, 1.L%>`^n) = n
proof
  let L be domRing, x be Element of L;
  set r = <%-x, 1.L%>;
  defpred P[Nat] means degree BRoots (r`^$1) = $1;
A1: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat such that
A2: P[n];
    r`^(n+1) = (r`^n)*'r by POLYNOM5:19;
    then
A3: degree BRoots(r`^(n+1)) = degree BRoots (r`^n) + degree BRoots r by Lm2
      .= n + degree ({x},1)-bag by A2,Th51;
    card {x} = 1 by CARD_1:30;
    hence thesis by A3,Th10;
  end;
  len 1_. L = 1 & r`^0 = 1_. L by POLYNOM4:4,POLYNOM5:15;
  then
A4: P[ 0 ] by Th54;
  thus for n being Nat holds P[n] from NAT_1:sch 2(A4, A1);
end;
