
theorem Th32:
  for R being Abelian add-associative right_zeroed
  right_complementable associative distributive well-unital
commutative non trivial doubleLoopStr st R is Noetherian
  for n being Nat holds Polynom-Ring (n,R) is Noetherian
proof
  let R be Abelian add-associative right_zeroed right_complementable
  associative distributive well-unital commutative non trivial
  doubleLoopStr;
  assume
A1: R is Noetherian;
  defpred P[Nat] means Polynom-Ring($1,R) is Noetherian;
A2: now
    let k be Nat such that
A3: P[k];
    ex P being Function of Polynom-Ring(Polynom-Ring(k,R)), Polynom-Ring(k+
    1,R) st P is RingIsomorphism by Th31;
    hence P[k+1] by A3,Th27;
  end;
  ex P being Function of R, Polynom-Ring (0,R) st P is RingIsomorphism by Th28;
  then
A4: P[ 0 ] by A1,Th27;
  thus for n being Nat holds P[n] from NAT_1:sch 2(A4, A2);
end;
