
theorem Th26:
  for L being domRing, n being Element of NAT, p being Polynomial
  of L st p <> 0_. L holds p`^n <> 0_. L
proof
  let L be domRing, n be Element of NAT, p be Polynomial of L;
  defpred P[Nat] means p`^$1 <> 0_. L;
  assume
A1: p <> 0_. L;
A2: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat such that
A3: P[n];
    p`^(n+1) = (p`^n) *' p by POLYNOM5:19;
    hence thesis by A1,A3,Th18;
  end;
  (1_. L).0 = 1.L & (0_. L).0 = 0.L by FUNCOP_1:7,POLYNOM3:30;
  then
A4: P[ 0 ] by POLYNOM5:15;
  for n being Nat holds P[n] from NAT_1:sch 2(A4,A2);
  hence thesis;
end;
