
theorem Th23:
  for L be domRing for p be Polynomial of L st len p <> 0
   for n be Nat holds len(p`^n) = n*len p-n+1
proof
  let L be domRing;
  let p be Polynomial of L;
  defpred P[Nat] means len(p`^$1) = $1*len p-$1+1;
  assume
A1: len p <> 0;
A2: now
    len p >= 0+1 by A1,NAT_1:13;
    then len p-1 >= 0+1-1;
    then
A3: len p-'1 = len p-1 by XREAL_0:def 2;
A4: p.(len p -'1) <> 0.L by A1,Lm2;
    let n be Nat;
    assume
A5: P[n];
    n*(len p-'1)+1 >= 0+1 by XREAL_1:6;
    then (p`^n).(len (p`^n)-'1) <> 0.L by A5,A3,Lm2;
    then
A6: (p`^n).(len (p`^n)-'1) * p.(len p -'1) <> 0.L by A4,VECTSP_2:def 1;
    len(p`^(n+1)) = len((p`^n)*'p) by Th19
      .= n*len p - n + 1 + len p - 1 by A5,A6,POLYNOM4:10
      .= (n+1)*len p - (n+1) + 1;
    hence P[n+1];
  end;
  len(p`^0) = len(1_.(L)) by Th15
    .= 0*len p-0+1 by POLYNOM4:4;
  then
A7: P[0];
  thus for n be Nat holds P[n] from NAT_1:sch 2(A7,A2);
end;
